Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Graph each function. Compare to the parent graph. State the domain
Document related concepts
System of polynomial equations wikipedia , lookup
Cubic function wikipedia , lookup
Elementary algebra wikipedia , lookup
Quadratic equation wikipedia , lookup
History of algebra wikipedia , lookup
Quartic function wikipedia , lookup
Median graph wikipedia , lookup
Transcript
and a reflection across the x-axis. Another
way to identify the stretch is to notice that the yvalues in the table are –5 times the corresponding yvalues for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≤ 0}.
10-1 Square Root Functions
Graph each function. Compare to the parent
graph. State the domain and range.
3. 1. SOLUTION: x
0
0.5
y
0
≈ 2.1
1
3
2
≈ 4.2
3
≈ 5.2
SOLUTION: x
0
0.5
y
0 ≈ 0.2
4
6
2
≈ 0.5
3
≈ 0.6
4
≈ 0.7
The parent function
is multiplied by a value
less than 1 and greater than 0, so the the graph is a
vertical compression of
. Another way to
identify the compression is to notice that the y-values
The parent function is multiplied by a value
greater than 1, so the graph is a vertical stretch of
. Another way to identify the stretch is to
notice that the y-values in the table are 3 times the
corresponding y-values for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ 0}.
2. 1
≈ 0.3
in the table are times the corresponding y-values
for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ 0}.
4. SOLUTION: x
0
0.5
y
0 ≈ –3.5
1
–5
2
≈ –7.1
3
≈ –8.7
4
–10
SOLUTION: x
0
0.5
y
0 ≈ –0.4
The parent function
is multiplied by a value
less than 1, so the graph is a vertical stretch of
and a reflection across the x-axis. Another
way to identify the stretch is to notice that the yvalues in the table are –5 times the corresponding yvalues for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≤ 0}.
3
≈ –0.9
2
≈ 0.5
3
≈ 0.6
4
≈ 0.7
4
–1
y-values in the table are
times the corresponding
y-values for the parent function. The domain is {x|x
≥ 0} and the range is {y|y ≤ 0}.
5. eSolutions Manual - Powered by Cognero
1
≈ 0.3
2
≈ –0.7
The parent function
is multiplied by a value
less than 1, so the graph is a vertical compression of
and a reflection across the x-axis. Another
way to identify the compression is to notice that the
3. SOLUTION: x
0
0.5
y
0 ≈ 0.2
1
–0.5
Page 1
SOLUTION: x
0
0.5
1
2
3
4
units from the parent graph
. Another way to
identify the translation is to note that the y-values in
the table are 2 less than the corresponding y-values
for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ –2}.
way to identify the compression is to notice that the
y-values in the table are
times the corresponding
10-1y-values
Square for
Root
theFunctions
parent function. The domain is {x|x
≥ 0} and the range is {y|y ≤ 0}.
5. 7. SOLUTION: x
0
0.5
y
3
≈ 3.7
1
4
2
≈ 4.4
3
≈ 4.7
SOLUTION: x
–2
y
0
4
5
–1
1
0
≈ 1.4
1
≈ 1.7
2
2
The value 3 is being added to the parent function
, so the graph is translated up 3 units from
The value 2 is being added to the square root of the
parent function
, so the graph is translated 2
the parent graph
. Another way to identify
the translation is to note that the y-values in the table
are 3 greater than the corresponding y-values for the
parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ 3}.
units left from the parent graph
. Another
way to identify the translation is to note that the xvalues in the table are 2 less than the corresponding
x-values for the parent function. The domain is {x|x
≥ –2}, and the range is {y|y ≥ 0}.
6. 8. SOLUTION: x
0
0.5
y
–2
≈ –
1.3
1
–1
2
≈ –
0.6
3
≈ –
0.3
4
0
The value 2 is being subtracted from the parent
function
, so the graph is translated down 2
units from the parent graph
. Another way to
identify the translation is to note that the y-values in
the table are 2 less than the corresponding y-values
for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ –2}.
4
1
5
≈ 1.4
6
≈ 1.7
7
2
The value 3 is being subtracted from the square root
of the parent function
, so the graph is
translated 3 units right from the parent graph
. Another way to identify the translation is to
note that the x-values in the table are 3 more than the
corresponding x-values for the parent function. The
domain is {x|x ≥ 3}, and the range is {y|y ≥ 0}.
9. FREE FALL The time t, in seconds, that it takes an
object to fall a distance d, in feet, is given by the
7. SOLUTION: eSolutions
- Powered–1
by Cognero0
x Manual–2
y
0
1
≈ 1.4
SOLUTION: x
3
y
0
function
1
≈ 1.7
2
2
(assuming zero air resistance). Page 2
Graph the function, and state the domain and range.
SOLUTION: translated 3 units right from the parent graph
. Another way to identify the translation is to
note that the x-values in the table are 3 more than the
10-1corresponding
Square Rootx-values
Functions
for the parent function. The
domain is {x|x ≥ 3}, and the range is {y|y ≥ 0}.
The domain is {d | d ≥ 0} and the range is {t | t ≥ 0}.
9. FREE FALL The time t, in seconds, that it takes an
object to fall a distance d, in feet, is given by the
Graph each function, and compare to the parent
graph. State the domain and range.
function
(assuming zero air resistance). 10. Graph the function, and state the domain and range.
SOLUTION: x
0
0.5
y
2
≈ 2.4
SOLUTION: Let d = 0, 1, 4, 9, and 16 feet. Make a table to find
accompanying values for t.
d
t
0
0
1
2.5
2
≈ 2.7
3
≈ 2.9
4
3
1
4
9
16
The parent function is multiplied by a value
less than 1 and is added to the value 2, so the graph
is a vertical compression of
followed by a
translation 2 units up. The domain is {x|x ≥ 0}, and the range is {y|y ≥ 2}.
1
Plot each point (d, t) and connect them with a
smooth curve.
11. SOLUTION: x
0
0.5
y
–1 ≈ –
0.8
1
–1.25
2
≈ –
1.35
3
≈ –
1.4
4
–1.5
The domain is {d | d ≥ 0} and the range is {t | t ≥ 0}.
Graph each function, and compare to the parent
graph. State the domain and range.
The graph is the result of a vertical compression of
the graph of y =
followed by a reflection across
the x-axis, and then a translation 1 unit down. The
domain is {x|x ≥ 0}, and the range is {y|y ≤ –1}.
10. SOLUTION: x
0
0.5
y
2
≈ 2.4
1
2.5
eSolutions Manual - Powered by Cognero
2
≈ 2.7
3
≈ 2.9
4
3
12. SOLUTION: x
–1
y
0
0
–2
1
≈ –2.8
2
≈ –3.5
3
Page 3
–4
The graph is the result of a vertical compression of
This graph is the result of a vertical stretch of the
the graph of y =
followed by a reflection across
10-1the
Square
Functions
x-axis,Root
and then
a translation 1 unit down. The
domain is {x|x ≥ 0}, and the range is {y|y ≤ –1}.
graph of y =
followed by a translation 2 units
right. The domain is {x|x ≥ 2}, and the range is {y|y
≥ 0}.
Graph each function. Compare to the parent
graph. State the domain and range.
12. SOLUTION: x
–1
y
0
14. 0
–2
1
≈ –2.8
2
≈ –3.5
3
–4
SOLUTION: x
0
0.5
y
0
≈ 3.5
This graph is the result of a vertical stretch of the
graph of
followed by a reflection across the
x-axis, and then a translation 1 unit left. The domain
is {x|x ≥ –1}, and the range is {y|y ≤ 0}.
3
3
4
≈ 4.2
5
≈ 5.2
2
≈ 7.1
3
≈ 8.7
4
10
The parent function is multiplied by a value
greater than 1, so the graph is a vertical stretch of
. Another way to identify the stretch is to
notice that the y-values in the table are 5 times the
corresponding y-values for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ 0}.
13. SOLUTION: x
2
y
0
1
5
6
6
15. SOLUTION: x
0
0.5
y
0 ≈ 0.35
1
0.5
2
≈ 0.7
3
≈ 0.9
4
1
This graph is the result of a vertical stretch of the
graph of y =
followed by a translation 2 units
right. The domain is {x|x ≥ 2}, and the range is {y|y
≥ 0}.
Graph each function. Compare to the parent
graph. State the domain and range.
The parent function
is multiplied by a value
less than 1 and greater than 0, so the the graph is a
vertical compression of
. Another way to
identify the compression is to notice that the y-values
14. SOLUTION: x
0
0.5
y
0
≈ 3.5
1
5
eSolutions Manual - Powered by Cognero
2
≈ 7.1
3
≈ 8.7
4
10
in the table are times the corresponding y-values
for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ 0}.
16. Page 4
SOLUTION: greater than 1, so the graph is a vertical stretch of
. Another way to identify the stretch is to
notice that the y-values in the table are 7 times the
corresponding y-values for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ 0}.
identify the compression is to notice that the y-values
in the table are times the corresponding y-values
10-1for
Square
Rootfunction.
Functions
the parent
The domain is {x|x ≥ 0} and the range is {y|y ≥ 0}.
16. 18. SOLUTION: x
0
0.5
y
0
≈ –
0.2
1
–0.3
2
≈ –
0.5
3
≈ –
0.6
SOLUTION: x
0
0.5
y
0
≈ –
0.2
4
≈ –
0.7
1
–
0.25
2
≈ –
0.35
3
≈ –
0.4
4
–0.5
The parent function
is multiplied by a value
less than 1, so the graph is a vertical compression of
and a reflection across the x-axis. Another
way to identify the compression is to notice that the
The parent function
is multiplied by a value
less than 1, so the graph is a vertical compression of
and a reflection across the x-axis. Another
way to identify the compression is to notice that the
y-values in the table are
times the corresponding
y-values for the parent function. The domain is {x|x
≥ 0} and the range is {y|y ≤ 0}.
y-values in the table are
times the corresponding
y-values for the parent function. The domain is {x|x
≥ 0} and the range is {y|y ≤ 0}.
17. 19. SOLUTION: x
0
0.5
y
0
≈ 4.9
1
7
2
≈ 9.9
3
≈ 12.1
SOLUTION: x
0
0.5
y
0
≈ –
0.7
4
14
The parent function is multiplied by a value
greater than 1, so the graph is a vertical stretch of
. Another way to identify the stretch is to
notice that the y-values in the table are 7 times the
corresponding y-values for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ 0}.
1
–1
The parent function
2
≈ –
1.4
3
≈ –
1.7
4
–2
is multiplied by the
negative of 1, so the graph is a reflection of across the x-axis. Another way to identify the graph
is to notice that the y-values in the table are –1 times
the corresponding y-values for the parent function.
The domain is {x|x ≥ 0} and the range is {y|y ≤ 0}.
18. 20. eSolutions Manual - Powered by Cognero
SOLUTION: x
0
0.5
y
0
≈ –
1
–
Page 5
2
≈ –
3
≈ –
4
–0.5
SOLUTION: x
0
0.5
1
2
3
4
negative of 1, so the graph is a reflection of across the x-axis. Another way to identify the graph
is to notice that the y-values in the table are –1 times
10-1the
Square
Root Functions
corresponding
y-values for the parent function.
The domain is {x|x ≥ 0} and the range is {y|y ≤ 0}.
and a reflection across the x-axis. Another
way to identify the stretch is to notice that the yvalues in the table are –7 times the corresponding yvalues for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≤ 0}.
22. 20. SOLUTION: x
0
0.5
y
0
≈ –
0.1
1
–0.2
2
≈ –
0.3
3
≈ –
0.35
SOLUTION: x
0
0.5
y
2
≈ 2.7
4
–0.4
1
3
2
≈ 3.4
3
≈ 3.7
4
4
The value 2 is being added to the parent function
, so the graph is translated up 2 units from
The parent function
is multiplied by a value
less than 1, so the graph is a vertical compression of
and a reflection across the x-axis. Another
way to identify the compression is to notice that the
y-values in the table are
times the corresponding
y-values for the parent function. The domain is {x|x
≥ 0} and the range is {y|y ≤ 0}.
the parent graph
. Another way to identify
the translation is to note that the y-values in the table
are 2 greater than the corresponding y-values for the
parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ 2}.
23. SOLUTION: x
0
0.5
y
4
≈ 4.7
21. SOLUTION: x
0
0.5
y
0
≈ –
4.9
1
–7
2
≈ –
9.9
3
≈ –
12.1
1
5
2
≈ 5.4
3
≈ 5.7
4
6
4
–14
The value 4 is being added to the parent function
, so the graph is translated up 4 units from
The parent function
is multiplied by a value
less than 1, so the graph is a vertical stretch of
and a reflection across the x-axis. Another
way to identify the stretch is to notice that the yvalues in the table are –7 times the corresponding yvalues for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≤ 0}.
eSolutions
Manual - Powered by Cognero
22. SOLUTION: the parent graph
. Another way to identify
the translation is to note that the y-values in the table
are 4 greater than the corresponding y-values for the
parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ 4}.
24. SOLUTION: x
0
0.5
y
–1
≈ –
0.3
1
0
2
≈ 0.4
3
≈ 0.7
4
Page
1 6
the parent graph
. Another way to identify
the translation is to note that the y-values in the table
are 4 greater than the corresponding y-values for the
10-1parent
Square
Root Functions
function.
The domain is {x|x ≥ 0} and the range is {y|y ≥ 4}.
24. units from the parent graph
. Another way to
identify the translation is to note that the y-values in
the table are 3 less than the corresponding y-values
for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ –3}.
26. SOLUTION: x
0
0.5
y
–1
≈ –
0.3
1
0
2
≈ 0.4
3
≈ 0.7
SOLUTION: x
0
0.5
y
1.5 ≈ 2.2
4
1
1
2.5
2
≈ 2.9
3
≈ 3.2
4
3.5
The value 1.5 is being added to the parent function
, so the graph is translated up 1.5 units from
The value 1 is being subtracted from the parent
function
, so the graph is translated down 1
the parent graph
. Another way to identify
the translation is to note that the y-values in the table
are 1.5 greater than the corresponding y-values for
the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ 1.5}.
unit from the parent graph
. Another way to
identify the translation is to note that the y-values in
the table are 1 less than the corresponding y-values
for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ –1}.
27. 25. SOLUTION: x
0
0.5
y
–3
≈ –
2.3
1
–2
2
≈ –
1.6
3
≈ –
1.3
SOLUTION: x
0
1
y
–2.5 –1.5
4
–1
2
–1.1
4
–0.5
6
≈ –
0.1
6.5
≈ 0
The value 2.5 is being subtracted from the parent
function
, so the graph is translated down 2.5
The value 3 is being subtracted from the parent
function
, so the graph is translated down 3
units from the parent graph
. Another way to
identify the translation is to note that the y-values in
the table are 2.5 less than the corresponding y-values
for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ –2.5}.
units from the parent graph
. Another way to
identify the translation is to note that the y-values in
the table are 3 less than the corresponding y-values
for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ –3}.
28. 26. eSolutions
Manual - Powered by Cognero
SOLUTION: x
y
0
1.5
0.5
≈ 2.2
1
2.5
2
≈ 2.9
3
≈ 3.2
4
3.5
SOLUTION: x
–4
–3
y
0
1
–2
≈ 1.4
0
2
2
≈ 2.4
Page
4 7
≈ 2.8
translated 4 units right from the parent graph
. Another way to identify the translation is to
note that the x-values in the table are 4 more than the
corresponding x-values for the parent function. The
domain is {x|x ≥ 4}, and the range is {y|y ≥ 0}.
units from the parent graph
. Another way to
identify the translation is to note that the y-values in
the table are 2.5 less than the corresponding y-values
10-1for
Square
Rootfunction.
Functions
the parent
The domain is {x|x ≥ 0} and the range is {y|y ≥ –2.5}.
28. 30. SOLUTION: x
–4
–3
y
0
1
–2
≈ 1.4
0
2
2
≈ 2.4
SOLUTION: x
–1
0
y
0
1
4
≈ 2.8
1
≈ 1.4
2
≈ 1.7
3
2
5
≈ 2.4
The value 4 is being added to the square root of the
parent function
, so the graph is translated 4
The value 1 is being added to the square root of the
parent function
, so the graph is translated 1
units left from the parent graph
. Another
way to identify the translation is to note that the xvalues in the table are 4 less than the corresponding
x-values for the parent function. The domain is {x|x
≥ –4}, and the range is {y|y ≥ 0}.
unit left from the parent graph
. Another way
to identify the translation is to note that the x-values
in the table are 1 less than the corresponding xvalues for the parent function. The domain is {x|x ≥ –1}, and the range is {y|y ≥ 0}.
29. 31. SOLUTION: x
4
5
y
0
1
6
≈ 1.4
6.5
≈ 1.6
7
≈ 1.7
SOLUTION: x
0.5
1
y
0
≈ 0.7
8
2
1.5
1
2
≈ 1.2
3
≈ 1.6
4.5
2
The value 0.5 is being subtracted from the square
root of the parent function
, so the graph is
translated 0.5 unit right from the parent graph
. Another way to identify the translation is to
note that the x-values in the table are 0.5 more than
the corresponding x-values for the parent function.
The domain is {x|x ≥ 0.5}, and the range is {y|y ≥ 0}.
The value 4 is being subtracted from the square root
of the parent function
, so the graph is
translated 4 units right from the parent graph
. Another way to identify the translation is to
note that the x-values in the table are 4 more than the
corresponding x-values for the parent function. The
domain is {x|x ≥ 4}, and the range is {y|y ≥ 0}.
32. 30. eSolutions
Manual - Powered by Cognero
SOLUTION: x
y
–1
0
0
1
1
≈ 2
≈ 1.7
3
2
5
≈ SOLUTION: x
–5
–4
y
0
1
–1
2
0
≈ 2.2
2
≈ 2.6
Page 8
4
3
translated 0.5 unit right from the parent graph
. Another way to identify the translation is to
note that the x-values in the table are 0.5 more than
10-1the
Square
Root Functions
corresponding
x-values for the parent function.
The domain is {x|x ≥ 0.5}, and the range is {y|y ≥ 0}.
34. GEOMETRY The perimeter of a square is given
by the function
, where A is the area of the
square.
a. Graph the function.
b. Determine the perimeter of a square with an area
32. SOLUTION: x
–5
–4
y
0
1
note that the x-values in the table are 1.5 more than
the corresponding x-values for the parent function.
The domain is {x|x ≥ 1.5}, and the range is {y|y ≥ 0}.
–1
2
0
≈ 2.2
2
≈ 2.6
4
3
2
of 225 m .
c. When will the perimeter and the area be the same
value?
SOLUTION: a. Make a table of values. Use perfect squares for A
to make
easy to calculate.
A
4
9
16
25
P
8
12
16
20
The value 5 is being added to the square root of the
parent function
, so the graph is translated 5
units left from the parent graph
. Another
way to identify the translation is to note that the xvalues in the table are 5 less than the corresponding
x-values for the parent function. The domain is {x|x
≥ –5}, and the range is {y|y ≥ 0}.
33. b.
SOLUTION: x
1.5
2.5
y
0
1
3
≈ 1.2
4
≈ 1.6
5
≈ 1.9
5.5
2
The perimeter of the square would be 60m.
c. Set A = P, and substitute P for A. Then solve for
P.
The value 1.5 is being subtracted from the square
root of the parent function
, so the graph is
translated 1.5 units right from the parent graph
. Another way to identify the translation is to
note that the x-values in the table are 1.5 more than
the corresponding x-values for the parent function.
The domain is {x|x ≥ 1.5}, and the range is {y|y ≥ 0}.
34. GEOMETRY The perimeter of a square is given
by the function
, where A is the area of the
square.
eSolutions Manual - Powered by Cognero
a. Graph the function.
b. Determine the perimeter of a square with an area
2
The perimeter and area would be the same value
2
when the area is 16 m , the perimeter is 16 m, or the
length of the sides of the square are 4 m.
Page 9
Graph each function, and compare to the parent
graph. State the domain and range.
This graph is the result of a vertical stretch of the
10-1 Square Root Functions
The perimeter and area would be the same value
2
when the area is 16 m , the perimeter is 16 m, or the
length of the sides of the square are 4 m.
graph of y =
followed by a reflection across the
x-axis, and then a translation 3 units down. The
domain is {x|x ≥ 0}, and the range is {y|y ≤ –3}.
37. Graph each function, and compare to the parent
graph. State the domain and range.
SOLUTION: x
–2
–1
y
0
0.5
35. SOLUTION: x
0
0.5
y
2
≈ 0.6
1
0
2
≈ –
0.8
3
≈ –
1.5
0
≈ 0.7
1
≈ 0.9
2
1
4
≈ 1.2
4
–2
This graph is the result of a vertical compression of
the graph of y =
followed by a translation 2 units
left. The domain is {x|x ≥ –2}, and the range is {y|y
≥ 0}.
This graph is the result of a vertical stretch of the
graph of y =
followed by a reflection across the
x-axis, and then a translation 2 units up. The domain
is {x|x ≥ 0}, and the range is {y|y ≤ 2}.
38. SOLUTION: x
1
2
y
0
–1
36. SOLUTION: x
0
0.5
y
–3
≈ –
5.1
1
–6
2
≈ –
7.2
3
≈ –
8.2
graph of y =
followed by a reflection across the
x-axis, and then a translation 3 units down. The
domain is {x|x ≥ 0}, and the range is {y|y ≤ –3}.
4
≈ –
1.7
5
–2
This graph is the result of a reflection across the xaxis of the graph of
followed by a translation
1 unit right. The domain is {x|x ≥ 1}, and the range is
{y|y ≤ 0}.
39. SOLUTION: x
1
2
y
2
2.25
37. 2
1
3.5
≈ –
1.6
4
–9
This graph is the result of a vertical stretch of the
SOLUTION: x
–2
–1
0
1
eSolutions Manual - Powered by Cognero
y
0
0.5
≈ ≈ 0.9
0.7
3
≈ –
1.4
4
≈ 1.2
3
≈ 2.35
4
≈ 2.4
5
2.5
10
2.75
Page 10
This graph is the result of a reflection across the xaxis of the graph of
followed by a translation
10-11Square
Root
unit right.
TheFunctions
domain is {x|x ≥ 1}, and the range is
{y|y ≤ 0}.
This graph is the result of a vertical compression of
the graph of
followed by a translation 1 unit
up and 2 units right. The domain is {x|x ≥ 2}, and the range is {y|y ≥ 1}.
41. ENERGY An object has kinetic energy when it is in
motion. The velocity in meters per second of an
object of mass m kilograms with an energy of E
39. SOLUTION: x
1
2
y
2
2.25
3
≈ 2.35
4
≈ 2.4
5
2.5
10
2.75
joules is given by the function
. Use a
graphing calculator to graph the function that
represents the velocity of a basketball with a mass of
0.6 kilogram.
SOLUTION: Use a graphing calculator to graph the function
where m = 0.6.
This graph is the result of a vertical compression of
the graph of
followed by a translation 2 units
up and 1 unit right. The domain is {x|x ≥ 1}, and the range is {y|y ≥ 2}.
40. SOLUTION: x
2
3
y
1
1.5
4
≈ 1.6
5
≈ 1.9
6
2
7
≈ 2.1
42. GEOMETRY The radius of a circle is given by
, where A is the area of the circle.
a. Graph the function.
b. Use a graphing calculator to determine the radius
2
of a circle that has an area of 27 in .
This graph is the result of a vertical compression of
the graph of
followed by a translation 1 unit
up and 2 units right. The domain is {x|x ≥ 2}, and the range is {y|y ≥ 1}.
41. ENERGY An object has kinetic energy when it is in
motion. The velocity in meters per second of an
object of mass m kilograms with an energy of E
joules is given by the function
SOLUTION: r
0
1
7
10
≈1.5
≈1.8
. Use a
graphing calculator to graph the function that
with a mass of
0.6 kilogram.
represents
velocity
of a basketball
eSolutions
Manual -the
Powered
by Cognero
SOLUTION: a.
A
0 3.14
Page 11
43. SPEED OF SOUND The speed of sound in air is
determined by the temperature of the air. The speed
c in meters per second is given by c =
10-1 Square Root Functions
331.5
42. GEOMETRY The radius of a circle is given by
, where A is the area of the circle.
a. Graph the function.
b. Use a graphing calculator to determine the radius
, where t is the temperature of the
air in degrees Celsius.
a. Use a graphing calculator to graph the function.
b. How fast does sound travel when the
temperature is 55°C?
c. How is the speed of sound affected when the
temperature increases to 65°C? Explain.
SOLUTION: a.
2
of a circle that has an area of 27 in .
SOLUTION: a.
A
0 3.14
r
0
1
7
10
≈1.5
≈1.8
b. b. Use a graphing calculator to graph
. [0, 1000] scl: 20 by [0, 1000] scl: 10
Use the value option from the CALC menu to find
the value or r at A = 27.
[-2, 28] scl: 3 by [-2, 8] scl: 1
When the temperature is 55°C, sound travels at about 363.3 m/s.
c. At A = 27, r = 2.9. The radius of a circle that has an
2
area of 27 in is about 2.9 in.
43. SPEED OF SOUND The speed of sound in air is
determined by the temperature of the air. The speed
c in meters per second is given by c =
331.5
, where t is the temperature of the
air inManual
degrees
Celsius.
eSolutions
- Powered
by Cognero
a. Use a graphing calculator to graph the function.
b. How fast does sound travel when the
[0, 1000] scl: 20 by [0, 1000] scl: 10
When t is 65°C, c is about 368.8 m/s, so this 10degree increase results in an increase in speed of
about 5.5 m/s.
Page 12
44. MULTIPLE REPRESENTATIONS In this
problem, you will explore the relationship between
1000]Root
scl: 20
by [0, 1000] scl: 10
10-1[0,
Square
Functions
When t is 65°C, c is about 368.8 m/s, so this 10degree increase results in an increase in speed of
about 5.5 m/s.
44. MULTIPLE REPRESENTATIONS In this
problem, you will explore the relationship between
the graphs of square root functions and parabolas.
2
a. GRAPHICAL Graph y = x on a coordinate
system.
b. ALGEBRAIC Write a piecewise-defined
2
function to describe the graph of y = x in each
quadrant.
c. GRAPHICAL On the same coordinate system,
graph
.
and d. GRAPHICAL On the same coordinate system, graph the line y = x. Plot the points (2, 4), (4, 2), and
(1, 1).
e. ANALYTICAL Compare the graph of the
parabola to the graphs of the square root functions.
SOLUTION: a.
d.
e . The combined graphs of the square root functions
are the same size and shape as the parabola. They
are a reflection of the parabola across the line y = x.
Point (4, 2) is reflected to (2, 4). The point (1, 1) is on
both functions.
CHALLENGE Determine whether each statement is true or false . Provide an example
or counterexample to support your answer.
45. Numbers in the domain of a radical function will
always be nonnegative.
SOLUTION: Numbers in the domain of a radical function will not
always be nonnegative.For example the domain of
is {x|x ≥ −3}.
b. In the first quadrant, the function is
the second quadrant, the function is
Combine the two to write a piecewise-defined
function
. In
.
46. Numbers in the range of a radical function will
always be nonnegative.
SOLUTION: Numbers in the range of a radical function will not
always be nonnegative.For example, −6 and −5 are in
the range of
.
47. WRITING IN MATH Why are there limitations on
the domain and range of square root functions?
c.
SOLUTION: is not a real number. d.
eSolutions Manual - Powered by Cognero
The term underneath the square root cannot be
negative, which means the domain must be
restricted. The square root function will also never
attain a value less than 0, which means that the range
is restricted.
Page 13
48. CCSS TOOLS Write a radical function with a
domain of all real numbers greater than or equal to 2
SOLUTION: Numbers in the range of a radical function will not
always be nonnegative.For example, −6 and −5 are in
10-1 Square Root Functions
the range of
.
47. WRITING IN MATH Why are there limitations on
the domain and range of square root functions?
be all real numbers greater than or equal to 2
because there cannot be a negative number under
the radicand. The range is all real numbers less than
or equal to 5 because the function is translated 5
units up, but then reflected across the x-axis.
49. WHICH DOES NOT BELONG? Identify the
equation that does not belong. Explain.
SOLUTION: is not a real number. SOLUTION: does not belong because it is a translation
The term underneath the square root cannot be
negative, which means the domain must be
restricted. The square root function will also never
attain a value less than 0, which means that the range
is restricted.
of
. The other equations represent vertical
stretches or compressions since a each
is
multiplied by a constant.
48. CCSS TOOLS Write a radical function with a
domain of all real numbers greater than or equal to 2
and a range of all real numbers less than or equal to
5.
SOLUTION: Sample answer:
. The domain must
be all real numbers greater than or equal to 2
because there cannot be a negative number under
the radicand. The range is all real numbers less than
or equal to 5 because the function is translated 5
units up, but then reflected across the x-axis.
49. WHICH DOES NOT BELONG? Identify the
equation that does not belong. Explain.
[-2, 18] scl: 2 by [-2, 8] scl: 1
50. OPEN ENDED Write a function that is a
reflection, translation, and a dilation of the parent
graph
.
SOLUTION: Multiplying by –1 reflects the function. Multiplying by
3 dilates the function. Subtracting 1 from x translates
the function. SOLUTION: does not belong because it is a translation
of
. The other equations represent vertical
stretches or compressions since a each
is
multiplied by a constant.
[-2, 18] scl; 2 by [-1, 10] scl: 2
51. REASONING If the range of the function y =
a
is {y|y ≤ 0}, what can you conclude about the value of a? Explain your reasoning.
SOLUTION: The value of a is negative. For the function to have
negative y-values, the value of a must be negative.
For example has the range of {y|y ≤ 0}
eSolutions Manual - Powered by Cognero
[-2, 18] scl: 2 by [-2, 8] scl: 1
Page 14
52. WRITING IN MATH Compare and contrast the
graphs of f (x) =
and .
10-1 Square Root Functions
[-2, 18] scl; 2 by [-1, 10] scl: 2
51. REASONING If the range of the function y =
a
is {y|y ≤ 0}, what can you conclude about the value of a? Explain your reasoning.
53. SOLUTION: The value of a is negative. For the function to have
negative y-values, the value of a must be negative.
For example has the range of {y|y ≤ 0}
52. WRITING IN MATH Compare and contrast the
graphs of f (x) =
and .
SOLUTION: Both functions are translations of the square root
function. f (x) =
is a translation 2 units up. g
(x) =
is a translation 2 units to the left. When the constant is added or subtracted to x under
the radical, the function is translated left or right. If
the constant is added or subtracted outside the
radical, the square root function is translated up or
down. 53. Which function best represents the graph?
2
A y = x
B y = 2x
C y =
D y = x
SOLUTION: Choice B is an exponential function, which is not
represented by the graph.
Choice C is a square root function, which is not
represented by the graph.
Choice D is a linear function, which is not
represented by the graph.
Choice A is a quadratic function, which graphs as a
parabola.
The correct choice is A.
54. The statement “x < 10 and 3x − 2 ≥ 7” is true when
x is equal to
F 0
G 2
H 8
J 12
SOLUTION: Which function best represents the graph?
2
A y = x
B y = 2x
C y =
D y = x
SOLUTION: Choice B is an exponential function, which is not
represented by the graph.
Choice C is a square root function, which is not
represented by the graph.
Choice D is a linear function, which is not
eSolutions
Manual - Powered
by Cognero
represented
by the graph.
Choice A is a quadratic function, which graphs as a
parabola.
So, 3 ≤ x < 10. Neither 0 nor 2 are greater than 3,
and 12 is not less than 10. Eight is both greater than 3
and less than 10.
The correct choice is H.
55. Which of the following is the equation of a line
parallel to
and passing through (−2,
−1)?
A y =
B y = 2x + 3
C Page 15
D So, 3 ≤ x < 10. Neither 0 nor 2 are greater than 3,
and 12 is not less than 10. Eight is both greater than 3
less than
10-1and
Square
Root10.
Functions
The correct choice is H.
55. Which of the following is the equation of a line
parallel to
and passing through (−2,
Therefore, 13 of mulch are needed to cover the
flower beds.
Graph each function.
57. f (x) = | 3x + 2 |
SOLUTION: Since f (x) cannot be negative, the minimum point of
the graph is where f (x) = 0.
−1)?
A y =
B y = 2x + 3
C D Make a table of values
x
0
–2
−1
f (x)
4
1
2
SOLUTION: Parallel lines have the same slope. Substitute the
point and slope into the point-slope form of a linear
equation.
The correct choice is D.
1
5
2
8
58. f (x) =
56. SHORT RESPONSE A landscaper needs to
mulch 6 rectangular flower beds that are 8 feet by 4
feet and 4 circular flower beds each with a radius of
3 feet. One bag of mulch covers 25 square feet.
How many bags of mulch are needed to cover the
flower beds?
SOLUTION: This is a piecewise-defined function. Make a table of
values. Be sure to include the domain values for
which the function changes.
x
2
3
–3
−2
–1
f (x)
0
1
2
0
1
Notice that both functions are linear.
SOLUTION: 59. Therefore, 13 of mulch are needed to cover the
flower beds.
Graph each function.
57. f (x) = | 3x + 2 |
SOLUTION: Since
f (x) cannot
bebynegative,
eSolutions
Manual
- Powered
Cognero the minimum point of
the graph is where f (x) = 0.
SOLUTION: Make a table of values.
x
f (x)
–5
–4
–4
–3
–3
–2
–2
–1
0
–1
0
1
Page 16
10-1 Square Root Functions
59. 60. f (x) =
SOLUTION: Make a table of values.
x
f (x)
–5
–4
–4
–3
–3
–2
–2
–1
0
–1
0
1
1
2
2
3
3
4
4
5
SOLUTION: Since f (x) cannot be negative, the minimum point of
the graph is where f (x) = 0.
Make a table of values
x
0
2
–2
f (x)
1
4
0
6
60. f (x) =
SOLUTION: Since f (x) cannot be negative, the minimum point of
the graph is where f (x) = 0.
Graph each set of ordered pairs. Determine
whether the ordered pairs represent a linear
function, a quadratic function, or an exponential
function.
61. {(−2, 5), (−1, 3), (0, 1), (1, −1), (2, −3)}
SOLUTION: Make a table of values
x
0
2
–2
f (x)
1
4
0
6
The points can be connected by a straight line.
Thus, the ordered pairs represent a linear function.
62. {(0, 0), (1, 3), (2, 4), (3, 3), (4, 0)}
eSolutions Manual - Powered by Cognero
Graph each set of ordered pairs. Determine
whether the ordered pairs represent a linear
SOLUTION: Page 17
The points can be connected by a straight line.
Thus, the ordered pairs represent a linear function.
10-1 Square Root Functions
62. {(0, 0), (1, 3), (2, 4), (3, 3), (4, 0)}
A curve can be drawn with the given points. The
curve has no line of symmetry and is rapidly decreasing. Thus, the ordered pairs represent an
exponential function.
64. {(−4, 4), (−2, 1), (0, 0), (2, 1), (4, 4)}
SOLUTION: SOLUTION: A parabola can be drawn with the given points. The
curve has a line of symmetry and points are reflected
over this line. Thus, the ordered pairs represent a
quadratic function.
A parabola can be drawn with the given points. The
curve has a line of symmetry and points are reflected
over this line. Thus, the ordered pairs represent a
quadratic function.
63. SOLUTION: A curve can be drawn with the given points. The
curve has no line of symmetry and is rapidly decreasing. Thus, the ordered pairs represent an
exponential function.
64. {(−4, 4), (−2, 1), (0, 0), (2, 1), (4, 4)}
SOLUTION: 65. HEALTH Aida exercises every day by walking
and jogging at least 3 miles. Aida walks at a rate of 4
miles per hour and jogs at a rate of 8 miles per hour.
Suppose she has at most one half-hour to exercise
today.
a. Draw a graph showing the possible amounts of
time she can spend walking and jogging today.
b. List three possible solutions.
SOLUTION: a. Let x be the number of minutes Aida walks and y
be the number of minutes Aida jogs. If Aida can
spend no more than one-half hour exercising, then
the sum of the number of minutes she walks and jogs
must be less than or equal to 30. This is expressed by
the following inequality.
Since the time is represented in minutes, Aida's rate
walking and jogging must be converted from miles
per hour to miles per minute by dividing each by 60.
Her rate walking is
miles per minute and
her rate jogging is
miles per minute. The
following inequality that uses the distance formula (d
= rt), represents the fact that the distance Aida
walks plus the distance she jogs is greater than or
equal to a total distance of 3 miles.
A parabola can be drawn with the given points. The
curve has a line of symmetry and points are reflected
eSolutions
- Powered
Cognero
Thus,bythe
ordered pairs represent a
overManual
this line.
function.
quadratic
Page 18
her rate jogging is
miles per minute. The
following inequality that uses the distance formula (d
rt), represents
the fact that the distance Aida
10-1=Square
Root Functions
walks plus the distance she jogs is greater than or
equal to a total distance of 3 miles.
SOLUTION: The graph shows a positive correlation because as
the number of grams of fat increases, the amount of
Calories also increases.
Factor each monomial completely.
67. 28n
3
SOLUTION: Graph the two inequalities on the same coordinate
plane and find their intersection.
2
68. −33a b
SOLUTION: 69. 150rt
SOLUTION: 2 2
70. −378nq r
b. In order to walk and jog at least 3 miles and
exercise no more than 30 minutes, Aida could: walk
for 15 minutes and jog for 15 minutes; walk for 10
minutes and jog for 18 minutes; walk for 5 minutes
and jog for 23 minutes.
66. NUTRITION Determine whether the graph shows
a positive , negative, or no correlation. If there is a
positive or negative correlation, describe its meaning
in the situation.
SOLUTION: 3 2
71. 225a b c
SOLUTION: 2 4
72. −160x y
SOLUTION: SOLUTION: The graph shows a positive correlation because as
the number of grams of fat increases, the amount of
Calories also increases.
Factor each monomial completely.
3
eSolutions
67. 28n Manual - Powered by Cognero
SOLUTION: Page 19
10-2 Simplifying Radical Expressions
Simplify each expression.
6. 1. SOLUTION: SOLUTION: 7. SOLUTION: 2. SOLUTION: 8. SOLUTION: 3. SOLUTION: 9. SOLUTION: 4. SOLUTION: 10. MULTIPLE CHOICE Which expression is
equivalent to
5. ?
A SOLUTION: B C D 6. SOLUTION: eSolutions Manual - Powered by Cognero
7. SOLUTION: Page 1
SOLUTION: 10-2 Simplifying Radical Expressions
10. MULTIPLE CHOICE Which expression is
equivalent to
12. ?
SOLUTION: A B C D 13. SOLUTION: SOLUTION: 14. The correct choice is D.
SOLUTION: Simplify each expression.
11. SOLUTION: 15. SOLUTION: 12. SOLUTION: eSolutions Manual - Powered by Cognero
13. 16. Page 2
SOLUTION: 10-2 Simplifying Radical Expressions
20. 16. SOLUTION: SOLUTION: 21. SOLUTION: Simplify each expression.
17. SOLUTION: 22. SOLUTION: 18. SOLUTION: 23. SOLUTION: 19. SOLUTION: 24. SOLUTION: 20. SOLUTION: 25. SOLUTION: eSolutions Manual - Powered by Cognero
21. Page 3
26. SOLUTION: 10-2 Simplifying Radical Expressions
25. 30. SOLUTION: SOLUTION: 31. 26. SOLUTION: SOLUTION: 32. 27. SOLUTION: SOLUTION: 33. SOLUTION: 28. SOLUTION: 34. SOLUTION: 29. SOLUTION: 35. ROLLER COASTER Starting from a stationary
position, the velocity v of a roller coaster in feet per
second at the bottom of a hill can be approximated by
, where h is the height of the hill in feet.
30. SOLUTION: eSolutions Manual - Powered by Cognero
31. a. Simplify the equation.
b. Determine the velocity of a roller coaster at the
bottom of a 134-foot hill.
SOLUTION: a.
Page 4
SOLUTION: pump that is advertised to project water with a
velocity of 77 feet per second meet the fire
department’s need? Explain.
10-2 Simplifying Radical Expressions
SOLUTION: a.
35. ROLLER COASTER Starting from a stationary
position, the velocity v of a roller coaster in feet per
second at the bottom of a hill can be approximated by
, where h is the height of the hill in feet.
a. Simplify the equation.
b. Determine the velocity of a roller coaster at the
bottom of a 134-foot hill.
b. To determine the height of the water, substitute 70
for v in the function
SOLUTION: a.
.
A pump with a velocity of 70 feet per second will
pump water only to a maximum height of 76.6 feet.
Therefore, this pump will not meet the fire
department’s need.
c. To determine the height of the water, substitute 77
b. To determine the velocity of the roller coaster at
the bottom of the hill, substitute 134 for h in the
equation
.
for v in the function
.
The roller coaster will have a velocity of about 92.6
ft/sec at the bottom of a 134-foot hill.
36. CCSS PRECISION When fighting a fire, the
velocity v of water being pumped into the air is
A pump with a velocity of 77 feet per second will
pump water to a maximum height of 92.6 feet.
Therefore, this pump will meet the fire department’s
need.
modeled by the function
, where h
represents the maximum height of the water and g
2
represents the acceleration due to gravity (32 ft/s ).
a. Solve the function for h.
b. The Hollowville Fire Department needs a pump
that will propel water 80 feet into the air. Will a pump
advertised to project water with a velocity of 70 feet
per second meet their needs? Explain.
c. The Jackson Fire Department must purchase a
pump that will propel water 90 feet into the air. Will a
pump that is advertised to project water with a
velocity of 77 feet per second meet the fire
department’s need? Explain.
Simplify each expression.
37. SOLUTION: SOLUTION: a.
eSolutions Manual - Powered by Cognero
b. To determine the height of the water, substitute 70
Page 5
A pump with a velocity of 77 feet per second will
pump water to a maximum height of 92.6 feet.
10-2Therefore,
Simplifying
Expressions
thisRadical
pump will
meet the fire department’s
need.
Simplify each expression.
39. 37. SOLUTION: SOLUTION: 40. SOLUTION: 38. SOLUTION: 41. SOLUTION: 39. SOLUTION: 42. eSolutions Manual - Powered by Cognero
SOLUTION: Page 6
10-2 Simplifying Radical Expressions
42. 45. SOLUTION: SOLUTION: 46. 43. SOLUTION: SOLUTION: 47. SOLUTION: 44. SOLUTION: 48. SOLUTION: 45. SOLUTION: 49. ELECTRICITY The amount of current in amperes
I that an appliance uses can be calculated using the
eSolutions Manual - Powered by Cognero
Page 7
formula
, where P is the power in watts and
R is the resistance in ohms.
10-2 Simplifying Radical Expressions
An appliance uses about 3.9 amps of current if the
power used is 75 watts and the resistance is 5 ohms.
49. ELECTRICITY The amount of current in amperes
I that an appliance uses can be calculated using the
50. KINETIC ENERGY The speed v of a ball can be
determined by the equation
formula
, where P is the power in watts and
R is the resistance in ohms.
a. Simplify the formula.
b. How much current does an appliance use if the
power used is 75 watts and the resistance is 5 ohms?
SOLUTION: a.
, where k is the
kinetic energy and m is the mass of the ball.
a. Simplify the formula if the mass of the ball is 3
kilograms.
b. If the ball is traveling 7 meters per second, what
is the kinetic energy of the ball in Joules?
SOLUTION: a. If the mass of the ball is 3 kilograms, substitute m
= 3 into the equation.
b. Substitute P = 75 and R = 5 in the equation
.
b. Substitute V = 7 in the equation
.
An appliance uses about 3.9 amps of current if the
power used is 75 watts and the resistance is 5 ohms.
50. KINETIC ENERGY The speed v of a ball can be
determined by the equation
, where k is the
kinetic energy and m is the mass of the ball.
a. Simplify the formula if the mass of the ball is 3
kilograms.
b. If the ball is traveling 7 meters per second, what
is the kinetic energy of the ball in Joules?
SOLUTION: a. If the mass of the ball is 3 kilograms, substitute m
= 3 into the equation.
The kinetic energy of the ball is 73.5 Joules.
51. SUBMARINES The greatest distance d in miles th
the lookout can see on a clear day is modeled by the
formula
. Determine how high the
submarine would have to raise its periscope to see a
ship, if the submarine is the given distances away fro
a ship.
eSolutions Manual - Powered by Cognero
Page 8
=
d
(3)
2
10-2 Simplifying Radical Expressions
The kinetic energy of the ball is 73.5 Joules.
2
=6
=
24
(9)
2
=
54
2
(12)
= 96
2
(15)
= 150
51. SUBMARINES The greatest distance d in miles th
the lookout can see on a clear day is modeled by the
formula
(6)
52. CCSS STRUCTURE Explain how to solve
.
. Determine how high the
submarine would have to raise its periscope to see a
ship, if the submarine is the given distances away fro
a ship.
SOLUTION: To solve an equation of equal ratios, first find the
equal cross products and then solve for the variable.
Use the conjugate of denominator.
to rationalize the SOLUTION: First solve the equation for h.
So, the solution is
Distance
Height
=
d
3
h=
2
6
h=
(3)
2
=6
9
h=
(6)
2
=
24
(9)
2
=
54
12
15
h=
h=
2
(12)
= 96
2
(15)
= 150
52. CCSS STRUCTURE Explain how to solve
.
SOLUTION: To solve an equation of equal ratios, first find the
equal cross products and then solve for the variable.
eSolutions Manual - Powered by Cognero
.
53. CHALLENGE Simplify each expression.
a.
b.
c.
SOLUTION: a.
b.
c.
54. REASONING Marge takes a number, subtracts 4,
multiplies by 4, takes the square root, and takes the
reciprocal to get
. What number did she start
with? Write a formula to describe the process.
SOLUTION: Page 9
b.
10-2 Simplifying Radical Expressions
c.
are
54. REASONING Marge takes a number, subtracts 4,
multiplies by 4, takes the square root, and takes the
and .
56. CHALLENGE Use the Quotient Property of
Square Roots to derive the Quadratic Formula by
2
reciprocal to get
. What number did she start
with? Write a formula to describe the process.
solving the quadratic equation ax + bx + c = 0.
(Hint: Begin by completing the square.)
SOLUTION: SOLUTION: Let x = a number.
57. WRITING IN MATH Summarize how to write a
radical expression in simplest form.
55. OPEN ENDED Write two binomials of the form
and . Then find their
product.
SOLUTION: and Two binomials of the form
are
and SOLUTION: No radicals can appear in the denominator of a
fraction. So, rationalize the denominator to get rid of
the radicand in the denominator. Then check if any
of the radicands have perfect square factors other
than 1. If so, simplify.
For example, simplify the following.
.
56. CHALLENGE Use the Quotient Property of
Square Roots to derive the Quadratic Formula by
2
solving the quadratic equation ax + bx + c = 0.
(Hint: Begin by completing the square.)
SOLUTION: eSolutions Manual - Powered by Cognero
58. Jerry’s electric bill is $23 less than his natural gas bill.
The two bills are a total of $109. Which of the
following equations can be used to find the amount of
his natural gas bill?
A g + g = 109
B 23 + 2g = 109
C g − 23 = 109
D 2g − 23 = 109
SOLUTION: Let g = Jerry’s natural gas bill. Jerry’s electric bill is
$23 less than his natural gas bill, so the electric bill is
g – 23.
The two bills are a total of $109.
Page 10
10-2 Simplifying Radical Expressions
The roots are –4 and 6. So, the correct choice is H.
58. Jerry’s electric bill is $23 less than his natural gas bill.
The two bills are a total of $109. Which of the
following equations can be used to find the amount of
his natural gas bill?
A g + g = 109
B 23 + 2g = 109
C g − 23 = 109
D 2g − 23 = 109
60. The expression
the following?
A is equivalent to which of B C D SOLUTION: Let g = Jerry’s natural gas bill. Jerry’s electric bill is
$23 less than his natural gas bill, so the electric bill is
g – 23.
The two bills are a total of $109.
SOLUTION: So, the correct choice is D.
So, the correct choice is C.
2
59. Solve a − 2a + 1 = 25.
F −4, −6
G 4, −6
H −4, 6
J 4, 6
61. GRIDDED RESPONSE Miki earns $10 an hour
and 10% commission on sales. If Miki worked 38
hours and had a total sales of $1275 last week, how
much did she make?
SOLUTION: Miki’s earnings are $10 per hour and 10%
commission on sales.
SOLUTION: Solve for a.
Miki made $507.50 last week.
Graph each function. Compare to the parent
graph. State the domain and range.
62. The roots are –4 and 6. So, the correct choice is H.
60. The expression
the following?
A SOLUTION: x
0
1
y
1
–1
2
≈ 1.8
3
≈ 2.5
4
3
is equivalent to which of B C D SOLUTION: eSolutions Manual - Powered by Cognero
is multiplied by a value
The parent function greater than 1 and is subtracted by the value 1, so the
Page 11
graph is a vertical stretch of
followed by a
translation 1 unit down. The domain is {x|x ≥ 0}, and the range is {y|y ≥ -1}.
identify the compression is to notice that the y-values
in the table are times the corresponding y-values
for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ 0}.
10-2 Simplifying Radical Expressions
Miki made $507.50 last week.
Graph each function. Compare to the parent
graph. State the domain and range.
64. SOLUTION: x
–2
–1
y
0
2
62. SOLUTION: x
0
1
y
1
–1
2
≈ 1.8
3
≈ 2.5
0
≈ 2.8
1
≈ 3.5
2
4
4
3
This graph is the result of a vertical stretch of the
is multiplied by a value
The parent function greater than 1 and is subtracted by the value 1, so the
graph is a vertical stretch of
followed by a
translation 1 unit down. The domain is {x|x ≥ 0}, and the range is {y|y ≥ -1}.
graph of y =
followed by a translation 2 units
left. The domain is {x|x ≥ –2}, and the range is {y|y
≥ 0}.
65. SOLUTION: x
0
–1
y
0
–1
63. SOLUTION: x
0
1
y
0
2
≈ 0.7
3
≈0 .9
1
≈ –
1.4
2
≈ –
1.7
4
1
This graph is the result of a reflection across the xaxis of the graph of
followed by a translation
1 unit left. The domain is {x|x ≥ –1}, and the range is
{y|y ≤ 0}.
The parent function
is multiplied by a value
less than 1 and greater than 0, so the the graph is a
vertical compression of
. Another way to
identify the compression is to notice that the y-values
in the table are times the corresponding y-values
for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ 0}.
64. eSolutions Manual - Powered by Cognero
SOLUTION: x
–2
–1
0
66. SOLUTION: x
3
4
y
0
–3
5
≈ –
4.2
6
≈ –
5.2
Page 12
1
2
This graph is the result of a reflection across the xaxis of the graph of
followed by a translation
unit left. TheRadical
domain is
{x|x ≥ –1}, and the range is
10-21Simplifying
Expressions
{y|y ≤ 0}.
Determine the domain and range for each
function.
68. f (x) = |2x − 5|
66. SOLUTION: x
3
4
y
0
–3
less than 1 and is added to the value 1, so the graph
is a vertical stretch of
followed by a
reflection across the x-axis, and then a translation 1
unit up. The domain is {x|x ≥ 0}, and the range is {y|y ≤ 1}.
5
≈ –
4.2
6
≈ –
5.2
This graph is the result of a vertical stretch of the
graph of
followed by a reflection across the
x-axis, and then a translation 3 units right. The
domain is {x|x ≥ 3}, and the range is {y|y ≤ 0}.
SOLUTION: Since f (x) cannot be negative, the minimum point of
the graph is where f (x) = 0.
Make a table of values.
x
0
1
f (x)
5
3
2
1
3
1
4
3
67. SOLUTION: x
0
1
y
1
–1
2
≈ –
1.8
3
≈ –
2.5
The domain is all real numbers, and the range is {y |
y ≥ 0}.
69. h(x) = |x − 1|
SOLUTION: is multiplied by a value
The parent function less than 1 and is added to the value 1, so the graph
is a vertical stretch of
followed by a
reflection across the x-axis, and then a translation 1
unit up. The domain is {x|x ≥ 0}, and the range is {y|y ≤ 1}.
Determine the domain and range for each
function.
68. f (x) = |2x − 5|
SOLUTION: Since f (x) cannot be negative, the minimum point of
the graph is where f (x) = 0.
eSolutions Manual - Powered by Cognero
Since h(x) cannot be negative, the minimum point of
the graph is where h(x) = 0.
Make a table of values.
x
–4
–2
h(x)
5
3
0
1
2
1
4
3
Page 13
The domain is all real numbers, and the range is {y |
y ≥ 0}.
10-2 Simplifying Radical Expressions
69. h(x) = |x − 1|
The domain is all real numbers, and the range is {y |
y ≥ 0}.
70. SOLUTION: Since h(x) cannot be negative, the minimum point of
the graph is where h(x) = 0.
Make a table of values.
x
–4
–2
h(x)
5
3
0
1
2
1
4
3
The domain is all real numbers, and the range is {y |
y ≥ 0}.
SOLUTION: This is a piecewise-defined function. Make a table
of values. Be sure to include the domain values for
which the function changes.
x
-1
0
1
2
3
g(x)
-2
-1
0
1
-5
Notice that both functions are linear.
The domain is all real numbers, and the range is {y |
y ≤ 1}.
Solve each equation by using the Quadratic
Formula. Round to the nearest tenth if
necessary.
2
71. x − 25 = 0
70. SOLUTION: SOLUTION: For this equation, a = 1, b = 0, and c = –25.
This is a piecewise-defined function. Make a table
of values. Be sure to include the domain values for
which the function changes.
x
-1
0
1
2
3
g(x)
-2
-1
0
1
-5
Notice that both functions are linear.
The solutions are 5 and –5.
2
72. r + 25 = 0
eSolutions Manual - Powered by Cognero
The domain is all real numbers, and the range is {y |
SOLUTION: For this equation, a = 1, b = 0, and c = 25.
Page 14
10-2The
Simplifying
solutions Radical
are 5 andExpressions
–5.
2
72. r + 25 = 0
SOLUTION: For this equation, a = 1, b = 0, and c = 25.
The solution is 5.
2
74. 2r + r − 14 = 0
SOLUTION: For this equation, a = 2, b = 1, and c = –14.
There are no real positive square roots of –100.
Therefore, the solution to this equation is ø.
2
73. 4w + 100 = 40w
SOLUTION: Rewrite the equation in standard form.
The solutions are –2.9 and 2.4.
2
75. 5v − 7v = 1
SOLUTION: Rewrite the equation in standard form.
For this equation, a = 4, b = –40, and c = 100.
For this equation, a = 5, b = –7, and c = –1.
The solution is 5.
2
74. 2r + r − 14 = 0
SOLUTION: For this equation, a = 2, b = 1, and c = –14.
The solutions are −0.1 and 1.5.
2
76. 11z − z = 3
SOLUTION: Rewrite the equation in standard form.
For this equation, a = 11, b = –1, and c = –3.
eSolutions Manual - Powered by Cognero
Page 15
SOLUTION: 10-2 Simplifying Radical Expressions
The solutions are −0.1 and 1.5.
2
76. 11z − z = 3
SOLUTION: Rewrite the equation in standard form.
4
80. 32x − 2y
4
SOLUTION: For this equation, a = 11, b = –1, and c = –3.
2
81. 4t − 27
The solutions are −0.5 and 0.6.
Factor each polynomial, if possible. If the
polynomial cannot be factored, write prime .
2
77. n − 81
SOLUTION: SOLUTION: In this trinomial, a = 4, b = 0 and c = –27, so m + p is
zero and mp is negative. Therefore, m and p must be
opposite signs. List the factors of 4(–27) or –108
with a sum of 0.
Factors of –108
Sum of 0
1, –108
–107
107
–1, 108
2, –52
–50
50
–2, 52
3, –36
–33
33
–3, 36
4, –27
–23
23
–4, 27
6, –18
–12
12
–6, 18
9, –12
–3
3
–9, 12
2
78. 4 − 9a
There are no factors of –108 with a sum of 0, so 4t
− 27 is prime.
2
SOLUTION: 3
2
82. x − 3x − 9x + 27
SOLUTION: 5
79. 2x − 98x
3
SOLUTION: 4
80. 32x − 2y
4
SOLUTION: eSolutions Manual - Powered by Cognero
83. POPULATION The country of Latvia has been
experiencing a 1.1% annual decrease in population.
In 2009, its population was 2,261,294. If the trend
continues, predict Latvia’s population in 2019.
SOLUTION: Use the equation for exponential decay, with a =
2,261,294, r = 0.011, and t =10.
Page 16
86. 88
SOLUTION: 10-2 Simplifying Radical Expressions
83. POPULATION The country of Latvia has been
experiencing a 1.1% annual decrease in population.
In 2009, its population was 2,261,294. If the trend
continues, predict Latvia’s population in 2019.
SOLUTION: Use the equation for exponential decay, with a =
2,261,294, r = 0.011, and t =10.
87. 180
SOLUTION: 88. 31
SOLUTION: The number 31 is prime. So, the prime factorization
of 31 is 31.
89. 60
Latvia’s population in 2019 will be about 2,024,510
people.
84. TOMATOES There are more than 10,000 varieties
of tomatoes. One seed company produces seed
packages for 200 varieties of tomatoes. For how
many varieties do they not provide seeds?
SOLUTION: Let t be the number of varieties of tomatoes for
which the seed company does not produce. Since the
seed company produces packages for 200 varieties,
the total number of varieties can be expressed as: t +
200. Since there are more than 10,000 varieties of
tomatoes: t + 200 > 10,000.
Solving, we get: t > 10,000 - 200
t > 9,800
SOLUTION: 90. 90
SOLUTION: Write the prime factorization of each number.
85. 24
SOLUTION: 86. 88
SOLUTION: 87. 180
SOLUTION: eSolutions Manual - Powered by Cognero
88. 31
SOLUTION: Page 17
10-3 Operations with Radical Expressions
Simplify each expression.
7. 1. SOLUTION: SOLUTION: 2. SOLUTION: 8. SOLUTION: 3. SOLUTION: 9. SOLUTION: 4. SOLUTION: 5. SOLUTION: 10. SOLUTION: 6. SOLUTION: 11. SOLUTION: 7. 12. SOLUTION: eSolutions Manual - Powered by Cognero
SOLUTION: Page 1
SOLUTION: SOLUTION: 10-3 Operations with Radical Expressions
16. 12. SOLUTION: SOLUTION: 13. GEOMETRY The area A of a triangle can be
found by using the formula
, where b
represents the base and h is the height. What is the
area of the triangle shown?
17. SOLUTION: SOLUTION: 18. SOLUTION: 19. SOLUTION: The area of the triangle is
Simplify each expression.
.
20. 14. SOLUTION: SOLUTION: 15. SOLUTION: 21. SOLUTION: eSolutions Manual - Powered by Cognero
16. SOLUTION: Page 2
10-3 Operations with Radical Expressions
26. GEOMETRY Find the perimeter and area of a
21. and a length of
rectangle with a width of
SOLUTION: .
SOLUTION: 22. SOLUTION: The perimeter is
units.
23. SOLUTION: The area is 12 square units.
Simplify each expression.
24. 27. SOLUTION: SOLUTION: 25. SOLUTION: 28. SOLUTION: 26. GEOMETRY Find the perimeter and area of a
rectangle with a width of
and a length of
.
SOLUTION: eSolutions Manual - Powered by Cognero
Page 3
29. 10-3 Operations with Radical Expressions
32. 28. SOLUTION: SOLUTION: 33. ROLLER COASTERS The velocity v in feet per
second of a roller coaster at the bottom of a hill is
related to the vertical drop h in feet and the velocity
v0 of the coaster at the top of the hill by the formula
29. .
SOLUTION: a. What velocity must a coaster have at the top of a
225-foot hill to achieve a velocity of 120 feet per
second at the bottom?
b. Explain why v0 = v −
is not equivalent to
the formula given.
SOLUTION: a. Let v = 120 and h = 225.
30. SOLUTION: The coaster must have a velocity of 0 feet per
second at the top of the hill.
31. SOLUTION: b. Sample answer: In the formula given, we are
taking the square root of the difference, not the
square root of each term.
34. FINANCIAL LITERACY Tadi invests $225 in a
savings account. In two years, Tadi has $232 in his
account. You can use the formula
to find the average annual interest rate r that the
account has earned. The initial investment is v0 and
32. SOLUTION: eSolutions
Manual - Powered by Cognero
v2 is the amount in two years. What was the average
annual interest rate that Tadi’s account earned?
SOLUTION: Let v0 = 225 and let v2 = 232.
Page 4
second at the top of the hill.
b. Sample answer: In the formula given, we are
the square
of the Expressions
difference, not the
10-3taking
Operations
withroot
Radical
square root of each term.
34. FINANCIAL LITERACY Tadi invests $225 in a
savings account. In two years, Tadi has $232 in his
account. You can use the formula
The average annual interest rate that Tadi’s account
earned was about 1.5%.
35. ELECTRICITY Electricians can calculate the
electrical current in amps A by using the formula
to find the average annual interest rate r that the
account has earned. The initial investment is v0 and
v2 is the amount in two years. What was the average
annual interest rate that Tadi’s account earned?
SOLUTION: Let v0 = 225 and let v2 = 232.
, where w is the power in watts and r the
resistance in ohms. How much electrical current is
running through a microwave oven that has 850
watts of power and 5 ohms of resistance? Write the
number of amps in simplest radical form, and then
estimate the amount of current to the nearest tenth.
SOLUTION: Let w = 850 and let r = 5.
The average annual interest rate that Tadi’s account
earned was about 1.5%.
There are
or about 13 amps of electrical current running through the microwave oven.
35. ELECTRICITY Electricians can calculate the
electrical current in amps A by using the formula
, where w is the power in watts and r the
36. CHALLENGE Determine whether the following
statement is true or false . Provide a proof or
counterexample to support your answer.
x +y >
resistance in ohms. How much electrical current is
running through a microwave oven that has 850
watts of power and 5 ohms of resistance? Write the
number of amps in simplest radical form, and then
estimate the amount of current to the nearest tenth.
SOLUTION: Let w = 850 and let r = 5.
when x > 0 and y > 0
SOLUTION: Because x and y are both positive, each side of the
inequality represents a positive number. So, you can
prove the statement by squaring each side of the
inequality.
Because x > 0 and y > 0, the product 2xy must be
a positive number.Thus, 2xy > 0 is always true.
There are
or about 13 amps of electrical current running through the microwave oven.
eSolutions Manual - Powered by Cognero
36. CHALLENGE Determine whether the following
statement is true or false . Provide a proof or
Therefore,
for all x > 0 and y > 0.
is a true statement
Page 5
37. CCSS ARGUMENTS Make a conjecture about
are
or about 13 amps of electrical 10-3There
Operations
with
Radical Expressions
current running through the microwave oven.
36. CHALLENGE Determine whether the following
statement is true or false . Provide a proof or
counterexample to support your answer.
when x > 0 and y > 0
x +y >
SOLUTION: Because x and y are both positive, each side of the
inequality represents a positive number. So, you can
prove the statement by squaring each side of the
inequality.
Therefore,
for all x > 0 and y > 0.
is a true statement
37. CCSS ARGUMENTS Make a conjecture about
the sum of a rational number and an irrational
number. Is the sum rational or irrational? Is the
product of a nonzero rational number and an
irrational number rational or irrational? Explain your
reasoning.
SOLUTION: Examine the sum of several pairs of rational and
irrational numbers:
is in lowest terms, and is irrational.
is in lowest terms and is irrational. Because x > 0 and y > 0, the product 2xy must be
a positive number.Thus, 2xy > 0 is always true.
Therefore,
for all x > 0 and y > 0.
is a true statement
is irrational.
Examine the product of several pairs of non-zero
rational and irrational numbers:
is irrational
is irrational
37. CCSS ARGUMENTS Make a conjecture about
the sum of a rational number and an irrational
number. Is the sum rational or irrational? Is the
product of a nonzero rational number and an
irrational number rational or irrational? Explain your
reasoning.
SOLUTION: Examine the sum of several pairs of rational and
irrational numbers:
is in lowest terms, and is irrational.
is in lowest terms and is irrational. is irrational. From the above examples, we should come up with
the conjecture that the sum of a rational number and
an irrational number is irrational, and the product of a
rational number and an irrational number is irrational. 38. OPEN ENDED Write an equation that shows a
sum of two radicals with different radicands. Explain
how you could combine these terms.
SOLUTION: Sample answer:
is irrational.
Examine the product of several pairs of non-zero
rational and irrational numbers:
is irrational
When you simplify
eSolutions Manual - Powered by Cognero
is irrational
is irrational. , you get
. When you
simplify
, you get
. Because these two
numbers have the same radicand, you can add them.
Page 6
39. WRITING IN MATH Describe step by step how
to multiply two radical expressions, each with two
the conjecture that the sum of a rational number and
an irrational number is irrational, and the product of a
rational number and an irrational number is irrational. 10-3 Operations with Radical Expressions
38. OPEN ENDED Write an equation that shows a
sum of two radicals with different radicands. Explain
how you could combine these terms.
In 6 years, the population of the town will be 14,500.
41. GEOMETRY Which expression represents the
sum of the lengths of the 12 edges on this rectangular
solid?
SOLUTION: Sample answer:
A 2(a + b + c)
B 3(a + b + c)
When you simplify
, you get
. When you
simplify
, you get
. Because these two
numbers have the same radicand, you can add them.
C 4(a + b + c)
D 12(a + b + c)
39. WRITING IN MATH Describe step by step how
to multiply two radical expressions, each with two
terms. Write an example to demonstrate your
description.
SOLUTION: For example, you can use the FOIL method. You
multiply the first terms within the parentheses. Then
you multiply the outer terms within the parentheses.
Then you would multiply the inner terms within the
parentheses. And, then you would multiply the last
terms within each parentheses. Combine any like
terms and simplify any radicals. For example:
40. SHORT RESPONSE The population of a town is
13,000 and is increasing by about 250 people per
year. This can be represented by the equation p =
13,000 + 250y, where y is the number of years from
now and p represents the population. In how many
years will the population of the town be 14,500?
SOLUTION: Let p = 14,500.
SOLUTION: There are 4 edges that have a length of a, 4 edges
that have a length of b, and 4 edges that have a
length of c. So, the expression 4a + 4b + 4c or 4(a +
b + c) represents the sum of the lengths of the 12
edges of the rectangular solid. Choice C is the correct answer.
42. Evaluate
F 4; 4
G 4; 2
H 2; 4
J 2; 2
and
for n = 25.
SOLUTION: Substitute 25 for n in both expressions. Choice G is correct. 43. The current I in a simple electrical circuit is given by
the formula
In 6 years, the population of the town will be 14,500.
Which
expression
41. GEOMETRY eSolutions
Manual - Powered
by Cognero
represents the
sum of the lengths of the 12 edges on this rectangular
solid?
, where V is the voltage and R is
the resistance of the circuit. If the voltage remains
unchanged, what effect will doubling the resistance
of the circuit have on the current?
A The current will remain the same.
Page 7
10-3 Operations with Radical Expressions
Choice G is correct. 43. The current I in a simple electrical circuit is given by
the formula
45. , where V is the voltage and R is
SOLUTION: the resistance of the circuit. If the voltage remains
unchanged, what effect will doubling the resistance
of the circuit have on the current?
A The current will remain the same.
B The current will double its previous value.
46. C The current will be half its previous value.
SOLUTION: D The current will be two units more than its
previous value.
SOLUTION: The current and the resistance have an inverse
relationship. If the resistance is double and the
voltage remains the same, the current will be half of
its previous value. 47. Consider an example with the resistance is 4 and
voltage is 100. Find I when R doubles. SOLUTION: 48. SOLUTION: Choice C is the correct answer.
49. SOLUTION: Simplify.
44. SOLUTION: Graph each function. Compare to the parent
graph. State the domain and range.
50. eSolutions Manual - Powered by Cognero
45. SOLUTION: SOLUTION: x
0
y
0
0.5
≈ 1.4
1
2
2
≈ 2.8
Page 8
and a reflection across the x-axis. Another
way to identify the stretch is to notice that the yvalues in the table are –3 times the corresponding yvalues for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≤ 0}.
SOLUTION: 10-3 Operations with Radical Expressions
Graph each function. Compare to the parent
graph. State the domain and range.
52. SOLUTION: x
–1
–0.5
y
0
≈ 0.7
50. SOLUTION: x
0
y
0
0.5
≈ 1.4
1
2
0
1
1
≈ 1.4
2
≈ 1.7
3
≈ 2
2
≈ 2.8
The value 1 is being added to the square root of the
parent function
, so the graph is translated 1
is multiplied by a value
The parent function greater than 1, so the graph is a vertical stretch of
. Another way to identify the stretch is to
notice that the y-values in the table are 2 times the
corresponding y-values for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ 0}.
unit left from the parent graph
. Another way
to identify the translation is to note that the x-values
in the table are 1 less than the corresponding xvalues for the parent function. The domain is {x|x ≥ –1}, and the range is {y|y ≥ 0}.
53. 51. SOLUTION: x
0
y
0
0.5
≈ –2.1
1
–3
SOLUTION: x
4
4.5
y
0
≈ 0.7
2
≈ –4.2
5
1
6
≈ 1.4
7
≈ 1.7
8
2
The value 4 is being subtracted from the square root
of the parent function
, so the graph is
translated 4 units right from the parent graph
. Another way to identify the translation is to
note that the x-values in the table are 4 more than the
corresponding x-values for the parent function. The
domain is {x|x ≥ 4}, and the range is {y|y ≥ 0}.
The parent function
is multiplied by a value
less than 1, so the graph is a vertical stretch of
and a reflection across the x-axis. Another
way to identify the stretch is to notice that the yvalues in the table are –3 times the corresponding yvalues for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≤ 0}.
54. 52. SOLUTION: eSolutions Manual - Powered by Cognero
x
0
1
–1
–0.5
y
0
1
≈ 0.7
≈ 1.4
2
≈ 1.7
3
≈ 2
SOLUTION: x
0
0.5
y
3
≈ 3.7
1
4
2
≈ 4.4
3
≈ 4.7
4
5
Page 9
translated 4 units right from the parent graph
. Another way to identify the translation is to
note that the x-values in the table are 4 more than the
10-3corresponding
Operations with
Radical
Expressions
x-values
for the
parent function. The
domain is {x|x ≥ 4}, and the range is {y|y ≥ 0}.
units from the parent graph
. Another way to
identify the translation is to note that the y-values in
the table are 2 less than the corresponding y-values
for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ –2}.
Factor each trinomial.
2
56. x + 12x + 27
54. SOLUTION: x
0
0.5
y
3
≈ 3.7
1
4
2
≈ 4.4
3
≈ 4.7
4
5
SOLUTION: 2
57. y + 13y + 30
SOLUTION: 2
The value 3 is being added to the parent function
, so the graph is translated up 3 units from
the parent graph
. Another way to identify
the translation is to note that the y-values in the table
are 3 greater than the corresponding y-values for the
parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ 3}.
58. p − 17p + 72
SOLUTION: 2
59. x + 6x – 7
SOLUTION: 55. SOLUTION: x
0
0.5
y
–2
≈ –
1.3
1
–1
2
≈ –
0.6
3
≈ –
0.3
4
0
2
60. y − y − 42
SOLUTION: 2
61. −72 + 6w + w
SOLUTION: The value 2 is being subtracted from the parent
function
, so the graph is translated down 2
units from the parent graph
. Another way to
identify the translation is to note that the y-values in
the table are 2 less than the corresponding y-values
for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ –2}.
62. FINANCIAL LITERACY Determine the value of
an investment if $400 is invested at an interest rate of
7.25% compounded quarterly for 7 years.
SOLUTION: Use the formula for calculating compound interest.
Factor each trinomial.
2
56. x + 12x + 27
eSolutions
Manual - Powered by Cognero
SOLUTION: Page 10
SOLUTION: 10-3 Operations with Radical Expressions
62. FINANCIAL LITERACY Determine the value of
an investment if $400 is invested at an interest rate of
7.25% compounded quarterly for 7 years.
SOLUTION: Use the formula for calculating compound interest.
67. SOLUTION: 68. 3.6t + 6 − 2.5t = 8
The value of the investment after 7 years is about
$661.44.
SOLUTION: Solve each equation. Round each solution to the
nearest tenth, if necessary.
63. −4c − 1.2 = 0.8
SOLUTION: 64. −2.6q − 33.7 = 84.1
SOLUTION: 65. 0.3m + 4 = 9.6
SOLUTION: 66. SOLUTION: 67. eSolutions Manual - Powered by Cognero
SOLUTION: Page 11
10-4 Radical Equations
1. GEOMETRY The surface area of a basketball is x
square inches. What is the radius of the basketball if
the formula for the surface area of a sphere is SA =
3. SOLUTION: 2
4πr ?
SOLUTION: The surface area is x square inches, so substitute x
2
for SA in the formula SA = 4πr , then solve for r.
Check:
4. SOLUTION: So, the radius of the basketball is
inches.
Solve each equation. Check your solution.
2. SOLUTION: Check:
Check:
5. SOLUTION: 3. SOLUTION: eSolutions Manual - Powered by Cognero
Check:
Page 1
Because –1 does not satisfy the original equation, 3 is
the only solution.
10-4 Radical Equations
5. 7. SOLUTION: SOLUTION: Check:
Check:
Because 3 does not satisfy the original equation, 10 is
the only solution.
Because 3 does not satisfy the original equation, 6 is
the only solution.
8. EXERCISE Suppose the function
6. SOLUTION: Check:
,
where S represents speed in meters per second and
is the leg length of a person in meters, can
approximate the maximum speed that a person can
run.
a. What is the maximum running speed of a person
with a leg length of 1.1 meters to the nearest tenth of
a meter?
b. What is the leg length of a person with a running
speed of 6.7 meters per second to the nearest tenth
of a meter?
c. As leg length increases, does maximum speed
increase or decrease? Explain.
SOLUTION: a. Substitute 1.1 for
in the equation
.
Because –1 does not satisfy the original equation, 3 is
the only solution.
7. SOLUTION: The maximum running speed is about 8.2 meters per
second.
b. Substitute 6.7 for S in the equation
eSolutions
Manual - Powered by Cognero
Check:
.
Page 2
0.7
6.5
1.1
8.2
1.5
9.5
As the person’s leg length increases, the speed
increases.
10-4Because
Radical3Equations
does not satisfy the original equation, 6 is
the only solution.
8. EXERCISE Suppose the function
Solve each equation. Check your solution.
,
where S represents speed in meters per second and
is the leg length of a person in meters, can
approximate the maximum speed that a person can
run.
a. What is the maximum running speed of a person
with a leg length of 1.1 meters to the nearest tenth of
a meter?
b. What is the leg length of a person with a running
speed of 6.7 meters per second to the nearest tenth
of a meter?
c. As leg length increases, does maximum speed
increase or decrease? Explain.
SOLUTION: a. Substitute 1.1 for
in the equation
9. SOLUTION: Check:
.
10. SOLUTION: The maximum running speed is about 8.2 meters per
second.
b. Substitute 6.7 for S in the equation
.
Check:
11. SOLUTION: The leg length is about 0.7 meters.
c.
0.7
6.5
1.1
8.2
1.5
9.5
As the person’s leg length increases, the speed
increases.
Check:
Solve each equation. Check your solution.
eSolutions Manual - Powered by Cognero
Page 3
9. 12. 10-4 Radical Equations
11. 14. SOLUTION: SOLUTION: Check:
Check:
12. SOLUTION: 15. SOLUTION: Check:
Check:
13. SOLUTION: Because –4 does not satisfy the original equation, 3 is
the only solution.
Check:
16. SOLUTION: Check:
eSolutions
Manual - Powered by Cognero
14. SOLUTION: Page 4
10-4Because
Radical –4
Equations
does not satisfy the original equation, 3 is
the only solution.
16. Because 1 does not satisfy the original equation, 6 is
the only solution.
18. SOLUTION: SOLUTION: Check:
Check:
Because –2 does not satisfy the original equation, 3 is
the only solution.
17. Because –4 does not satisfy the original equation, 0 is
the only solution.
SOLUTION: 19. SOLUTION: Check:
Check:
Because 1 does not satisfy the original equation, 6 is
the only solution.
18. SOLUTION: The solution is 7.
20. SOLUTION: Check:
eSolutions Manual - Powered by Cognero
Page 5
10-4 Radical Equations
The solution is 7.
The solution is 47.
21. RIDES The amount of time t, in seconds, that it
takes a simple pendulum to complete a full swing is
20. SOLUTION: called the period . It is given by
, where
is the length of the pendulum, in feet.
a. The Giant Swing completes a period in about 8
seconds. About how long is the pendulum’s arm?
Round to the nearest foot.
b. Does increasing the length of the pendulum
increase or decrease the period? Explain.
Check:
SOLUTION: a. Substitute 8 for t in the equation
.
The solution is 47.
21. RIDES The amount of time t, in seconds, that it
takes a simple pendulum to complete a full swing is
called the period . It is given by
The length of the pendulum’s arm is about 52 feet.
b.
, where
is the length of the pendulum, in feet.
a. The Giant Swing completes a period in about 8
seconds. About how long is the pendulum’s arm?
Round to the nearest foot.
b. Does increasing the length of the pendulum
increase or decrease the period? Explain.
52
8.00
55
8.24
60
8.60
As the length of the pendulum increases, the period
also increases.
SOLUTION: a. Substitute 8 for t in the equation
Solve each equation. Check your solution.
.
22. SOLUTION: The length of the pendulum’s arm is about 52 feet.
b.
52
Use the quadratic equation to solve, with a = 1, b = –
8.00
eSolutions
- Powered by Cognero
55 Manual8.24
60
8.60
As the length of the pendulum increases, the period
Page 6
52
8.00
55
8.24
60
8.60
10-4As
Radical
Equations
the length
of the pendulum increases, the period
also increases.
This equation has no solution.
Solve each equation. Check your solution.
24. 22. SOLUTION: SOLUTION: Use the quadratic equation to solve, with a = 1, b = –
Check:
Since you cannot take the square root of a negative n
equation has no real solution.
23. SOLUTION: Check:
The solution is
.
25. SOLUTION: This equation has no solution.
24. SOLUTION: eSolutions Manual - Powered by Cognero
Page 7
Check:
Because –11 does not satisfy the original equation,
11 is the only solution.
10-4The
Radical
Equations
solution
is .
27. 25. SOLUTION: SOLUTION: Check:
Check:
Because –3 does not satisfy the original equation, 3 is
the only solution.
28. CCSS REASONING The formula for the slant
height c of a cone is
, where h is the
height of the cone and r is the radius of its base. Find
the height of the cone if the slant height is 4 and the
radius is 2. Round to the nearest tenth.
The solution is 235.2.
26. SOLUTION: SOLUTION: Substitute 4 for c and 2 for r in the equation
.
Check:
Because –11 does not satisfy the original equation,
11 is the only solution.
27. SOLUTION: eSolutions
Manual - Powered by Cognero
The height of the cone is
or ≈ 3.5.
Page 8
29. MULTIPLE REPRESENTATIONS Consider
a. GRAPHICAL Clear the Y= list. Enter the left s
10-4 Radical Equations
The height of the cone is
or ≈ 3.5.
Use the quadratic equation to solve, with a = 1, b = –
29. MULTIPLE REPRESENTATIONS Consider
a. GRAPHICAL Clear the Y= list. Enter the left s
x − 7. Press GRAPH.
b. GRAPHICAL Sketch what is shown on the scr
c. ANALYTICAL Use the intersect feature on t
d. ANALYTICAL Solve the radical equation alge
SOLUTION: a.
Check:
Use the WINDOW menu to adjust the viewing wi
Because 5.17 does not satisfy the original equation, 1
calculator.
[-10, 20] scl: 1 by [-10, 10] scl: 1
b. See students’ work.
c.
30. PACKAGING A cylindrical container of chocolate
drink mix has a volume of 162 cubic inches. The
radius r of the container can be found by using the
formula
, where V is the volume of the
container and h is the height.
a. If the radius is 2.5 inches, find the height of the
container. Round your answer to the nearest
hundredth.
b. If the height of the container is 10 inches, find the
radius. Round to the nearest hundredth.
[-10, 20] scl: 1 by [-10, 10] scl: 1 [-10, 20] scl: 1 by [
d.
SOLUTION: a. Substitute 162 for V and 2.5 for r in the equation
.
Use the quadratic equation to solve, with a = 1, b = –
eSolutions Manual - Powered by Cognero
Page 9
10-4Because
Radical5.17
Equations
does not satisfy the original equation, 1
calculator.
30. PACKAGING A cylindrical container of chocolate
drink mix has a volume of 162 cubic inches. The
radius r of the container can be found by using the
formula
The radius of the container is about 2.27 inches.
31. CCSS CRITIQUE Jada and Fina solved
. Is either of them correct?
Explain.
, where V is the volume of the
container and h is the height.
a. If the radius is 2.5 inches, find the height of the
container. Round your answer to the nearest
hundredth.
b. If the height of the container is 10 inches, find the
radius. Round to the nearest hundredth.
SOLUTION: SOLUTION: a. Substitute 162 for V and 2.5 for r in the equation
.
Jada is correct. Fina had the wrong sign for 2b in the
fourth step.
32. REASONING Which equation has the same
solution set as
? Explain.
A. B. 4 = x + 2
C. SOLUTION: The height of the container is about 8.25 inches.
b. Substitute 162 for V and 10 for h in the equation
.
The solution to the equation is 2.
In choice A,
.
Choice B is the result of squaring both sides of the
equation, and 4 = 2 + 2.
The radius of the container is about 2.27 inches.
31. CCSS CRITIQUE Jada and Fina solved
. Is either of them correct?
Explain.
eSolutions Manual - Powered by Cognero
In choice C,
.
So, the correct choice is B.
33. REASONING Explain how solving
different from solving
SOLUTION: .
is Page 10
equation, and 4 = 2 + 2.
In choice C,
.
10-4 Radical Equations
So, the correct choice is B.
The solution is 1.
33. REASONING Explain how solving
different from solving
is .
SOLUTION: In the first equation, you have to isolate the radical
first by subtracting 1 from each side. Then square
each side to find the value of x. In the second
equation, the radical is already isolated, so square
each side to start. Then subtract 1 from each side to
solve for x.
Solve
35. REASONING Is the following equation sometimes,
always or never true? Explain.
SOLUTION: Sometimes; the equation is true for x ≥ 2, but false for x < 2, because if x is less than 2 then the right
side of the equation will be negative, and a square
root can never be negative.
. When x = 3,
Solve .
when x = 2,
34. OPEN ENDED Write a radical equation with a
variable on each side. Then solve the equation.
SOLUTION: Sample answer:
When x = 1.
36. CHALLENGE Solve
.
SOLUTION: Check:
Check:
The solution is 1.
35. REASONING Is the following equation sometimes,
eSolutions
Manual
Powered
by Cognero
always
or -never
true?
Explain.
Page 11
10-4 Radical Equations
When x = 1.
36. CHALLENGE Solve
.
SOLUTION: Check:
37. WRITING IN MATH Write some general rules
about how to solve radical equations. Demonstrate
your rules by solving a radical equation.
SOLUTION: Start by using addition/subtraction to isolate the term
containing the radical on one side of the equation.
Then multiply/divide each side by a number to
change the coefficient of the radical to +1.Next,
square each side of the equation to eliminate the
radical. Solve the resulting equation using the most
appropriate method. Last, check the answers found
in the original equation to remove any extraneous
solutions.
37. WRITING IN MATH Write some general rules
about how to solve radical equations. Demonstrate
your rules by solving a radical equation.
SOLUTION: Start by using addition/subtraction to isolate the term
containing the radical on one side of the equation.
Then multiply/divide each side by a number to
change the coefficient of the radical to +1.Next,
square each side of the equation to eliminate the
radical. Solve the resulting equation using the most
appropriate method. Last, check the answers found
in the original equation to remove any extraneous
solutions.
Check:
There are no extraneous solutions. Therefore, the
solution is
eSolutions Manual - Powered by Cognero
Check:
.
38. SHORT RESPONSE Zach needs to drill a hole at
Page 12
A, B, C, D, and E on circle P.
There are no extraneous solutions. Therefore, the
10-4solution
RadicalisEquations
.
38. SHORT RESPONSE Zach needs to drill a hole at
A, B, C, D, and E on circle P.
The measure of ∠CPD is 62.5°.
39. Which expression is undefined when w = 3?
A B C If Zack drills holes so that m APE = 110º and the
other four angles are congruent, what is m CPD?
SOLUTION: The measure of the central angles in a circle sum to
360, so
.
Let x = the measure of each of the other 4 angles.
D SOLUTION: For A;
For B;
The measure of ∠CPD is 62.5°.
39. Which expression is undefined when w = 3?
A For C;
B C D For D;
SOLUTION: For A;
The expression in C is undefined when w = 3.
For B;
40. What is the slope of a line that is parallel to the line?
For C;
F −3
eSolutions Manual - Powered by Cognero
G Page 13
The slope of the line on the graph is
parallel to the line on the graph has the same slope.
So the correct choice is H.
10-4 Radical Equations
The expression in C is undefined when w = 3.
40. What is the slope of a line that is parallel to the line?
. A line that is
41. What are the solutions of
A 1, 6
B −1, −6
C 1
D 6
?
SOLUTION: F −3
G H J 3
Check:
SOLUTION: First find the slope of the line on the graph. Use the
two points (0, 2) and (3, 3).
Because 1 does not satisfy the original equation, 6 is
the only solution. So, the correct choice is D.
The slope of the line on the graph is
. A line that is
42. ELECTRICITY The voltage V required for a
circuit is given by
, where P is the power
in watts and R is the resistance in ohms. How many
more volts are needed to light a 100-watt light bulb
than a 75-watt light bulb if the resistance of both is
110 ohms?
parallel to the line on the graph has the same slope.
So the correct choice is H.
41. What are the solutions of
A 1, 6
B −1, −6
C 1
D 6
?
SOLUTION: First find the number of volts needed to light a 100watt light bulb. Substitute 100 for P and 110 for R in
the equation
.
SOLUTION: Next find the number of volts needed to light a 75watt light bulb. Substitute 75 for P and 110 for R in
the equation
.
Check:
So, 105 – 91, or about 14 more volts are needed to
light a 100-watt light bulb.
Simplify each expression.
eSolutions Manual - Powered by Cognero
Because 1 does not satisfy the original equation, 6 is
43. Page 14
105 – Equations
91, or about 14 more volts are needed to
10-4So,
Radical
light a 100-watt light bulb.
Simplify each expression.
47. 43. SOLUTION: SOLUTION: 44. SOLUTION: 48. SOLUTION: 45. SOLUTION: 49. PHYSICAL SCIENCE A projectile is shot straight
up from ground level. Its height h, in feet, after t
46. 2
SOLUTION: seconds is given by h = 96t − 16t . Find the value(s)
of t when h is 96 feet.
SOLUTION: Use the quadratic equation to solve, with a = –16, b
= 96, and c = –96.
47. eSolutions
Manual - Powered by Cognero
SOLUTION: Page 15
Therefore, the projectile will be at 96 feet on the way
up at 1.3 seconds, and will be at 96 feet on the way
down at 4.7 seconds.
10-4 Radical Equations
49. PHYSICAL SCIENCE A projectile is shot straight
up from ground level. Its height h, in feet, after t
2
seconds is given by h = 96t − 16t . Find the value(s)
of t when h is 96 feet.
SOLUTION: Use the quadratic equation to solve, with a = –16, b
= 96, and c = –96.
Factor each trinomial, if possible. If the
trinomial cannot be factored using integers,
write prime .
2
50. 2x + 7x + 5
SOLUTION: In this trinomial, a = 2, b = 7 and c = 5, so m + p is
positive and mp is positive. Therefore, m and p must
both be positive. List the factors of 2(5) or 10.
Identify the factors with a sum of 7.
Factors of 10
Sum
1, 10
11
2, 5
7
The correct factors are 2 and 5.
2
So, 2x + 7x + 5= (2x + 5) (x + 1).
Therefore, the projectile will be at 96 feet on the way
up at 1.3 seconds, and will be at 96 feet on the way
down at 4.7 seconds.
Factor each trinomial, if possible. If the
trinomial cannot be factored using integers,
write prime .
2
50. 2x + 7x + 5
SOLUTION: In this trinomial, a = 2, b = 7 and c = 5, so m + p is
positive and mp is positive. Therefore, m and p must
both be positive. List the factors of 2(5) or 10.
Identify the factors with a sum of 7.
Factors of 10
Sum
1, 10
11
2, 5
7
The correct factors are 2 and 5.
2
So, 2x + 7x + 5= (2x + 5) (x + 1).
eSolutions Manual - Powered by Cognero
2
51. 6p + 5p − 6
SOLUTION: 2
51. 6p + 5p − 6
SOLUTION: In this trinomial, a = 6, b = 5 and c = –6, so m + p is
positive and mp is negative. Therefore, m and p must
be opposite signs. List the factors of 6(–6) or –36
with a sum of 5.
Factors of –36
Sum 1, –36
–35
35
–1, 36
2, –18
–16
16
–2, 18
3, –12
–9
9
–3, 12
4, –9
–5
5
–4, 9
The correct factors are –4 and 9.
2
So, 6p + 5p − 6= (2p + 3) (3p − 2).
2
52. 5d + 6d − 8
SOLUTION: Page 16
2
10-4So,
Radical
2x + Equations
7x + 5= (2x + 5) (x + 1).
2
2
So, 6p + 5p − 6= (2p + 3) (3p − 2).
2
51. 6p + 5p − 6
52. 5d + 6d − 8
SOLUTION: In this trinomial, a = 6, b = 5 and c = –6, so m + p is
positive and mp is negative. Therefore, m and p must
be opposite signs. List the factors of 6(–6) or –36
with a sum of 5.
Factors of –36
Sum 1, –36
–35
35
–1, 36
2, –18
–16
16
–2, 18
3, –12
–9
9
–3, 12
4, –9
–5
5
–4, 9
The correct factors are –4 and 9.
2
So, 6p + 5p − 6= (2p + 3) (3p − 2).
SOLUTION: In this trinomial, a = 5, b = 6 and c = –8, so m + p is
positive and mp is negative. Therefore, m and p must
be opposite signs. List the factors of 5(–8) or –40
with a sum of 6.
Factors of –40
Sum
1, –40
–39
39
–1, 40
2, –20
–18
18
–2, 20
4, –10
–6
6
–4, 10
5, –8
–3
3
–5, 8
The correct factors are –4 and 10.
2
So, 5d + 6d − 8= (5d − 4) (d + 2).
2
2
53. 8k − 19k + 9
52. 5d + 6d − 8
SOLUTION: In this trinomial, a = 5, b = 6 and c = –8, so m + p is
positive and mp is negative. Therefore, m and p must
be opposite signs. List the factors of 5(–8) or –40
with a sum of 6.
Factors of –40
Sum
1, –40
–39
39
–1, 40
2, –20
–18
18
–2, 20
4, –10
–6
6
–4, 10
5, –8
–3
3
–5, 8
The correct factors are –4 and 10.
SOLUTION: In this trinomial, a = 8, b = –19 and c = 9, so m + p is
negative and mp is positive. Therefore, m and p must
both be negative. List the negative factors of 8(9) or
72 with a sum of –19.
Factors of 72
Sum
–1, –72
–73
–2, –36
–38
–3, –24
–27
–4, –18
–22
–6, –12
–18
–8, –9
–17
There are no negative factors of 72 with a sum of –
19.
2
So, 5d + 6d − 8 is prime.
2
54. 9g − 12g + 4
2
eSolutions
Manual
- Powered
by Cognero
So, 5d
+ 6d
− 8= (5d
− 4) (d
2
53. 8k − 19k + 9
+ 2).
SOLUTION: In this trinomial, a = 9, b = –12 and c = 4, so m + p is
negative and mp is positive. Therefore, m and p must
both be negative. List the negative factors of 9(4) or
36 with a sum of –12.
Page 17
Factors of 36
Sum
–1, –36
–37
–2, –18
–20
–6, –12
–18
–8, –9
–17
There are no negative factors of 72 with a sum of –
10-419.
Radical Equations
2
2
So, 5d + 6d − 8 is prime.
So, 2a − 9a − 18= (2a + 3)(a − 6).
2
Determine whether each expression is a
monomial. Write yes or no. Explain.
54. 9g − 12g + 4
SOLUTION: In this trinomial, a = 9, b = –12 and c = 4, so m + p is
negative and mp is positive. Therefore, m and p must
both be negative. List the negative factors of 9(4) or
36 with a sum of –12.
Factors of 36
Sum
–1, –36
–37
–2, –18
–20
–3, –12
–15
–4, –9
–13
–6, –6
–12
The correct factors are –6 and –6.
56. 12
SOLUTION: Yes; 12 is a real number and therefore a monomial.
3
57. 4x
SOLUTION: 3
Yes; 4x is the product of a number and three
variables.
58. a − 2b
SOLUTION: No; a – 2b shows subtraction, not multiplication
alone of numbers and variables.
59. 4n + 5p
SOLUTION: No; 4n + 5p shows addition, not multiplication alone
of numbers and variables.
2
So, 9g − 12g + 4= (3g − 2)(3g − 2).
2
55. 2a − 9a − 18
SOLUTION: In this trinomial, a = 2, b = –9 and c = –18, so m + p
is negative and mp is negative. Therefore, m and p
must be opposite signs. List the factors of 2(–18) or
–36 with a sum of –9.
Factors of –36
Sum
1, –36
–35
35
–1, 36
2, –18
–16
16
–2, 18
3, –12
–9
9
–3, 12
4, –9
–5
5
–4, 9
The correct factors are 3 and –12.
60. SOLUTION: No;
has a variable in the denominator.
61. SOLUTION: Yes;
is the product of a number, , and several variables.
Simplify.
2
62. 9
SOLUTION: 2
So, 2a − 9a − 18= (2a + 3)(a − 6).
Determine whether each expression is a
monomial. Write yes or no. Explain.
eSolutions Manual - Powered by Cognero
56. 12
6
63. 10
SOLUTION: Page 18
SOLUTION: 10-4 Radical Equations
6
63. 10
SOLUTION: 5
64. 4
SOLUTION: 65. (8v)
2
SOLUTION: 66. SOLUTION: 2 3
67. (10y )
SOLUTION: eSolutions Manual - Powered by Cognero
Page 19
10-5 The Pythagorean Theorem
Find each missing length. If necessary, round to
the nearest hundredth.
4. 1. SOLUTION: Use the Pythagorean Theorem, substituting 3 for a
and 4 for b.
SOLUTION: Use the Pythagorean Theorem, substituting 8 for a
and 12 for b.
5. BASEBALL A baseball diamond is a square. The
distance between consecutive bases is 90 feet.
2. SOLUTION: Use the Pythagorean Theorem, substituting 4 for a
and 21 for c.
3. SOLUTION: Use the Pythagorean Theorem, substituting 6 for b
and 19 for c.
a. How far does a catcher have to throw the ball
from home plate to second base?
b. How far does a third baseman throw the ball to
the first baseman from a point in the baseline 15 feet
from third to second base?
c. A base runner going from first to second base is
100 feet from home plate. How far is the runner
from second base?
SOLUTION: a. Use the Pythagorean Theorem, substituting 90 for
a and 90 for b.
The catcher has to throw the ball about 127 ft from
home plate to second base.
4. SOLUTION: eSolutions
Manual - Powered by Cognero
Use the Pythagorean Theorem, substituting 8 for a
and 12 for b.
b. The diagram below illustrates the throw made by
the third baseman.
Page 1
has to throw
the ball about 127 ft from
10-5The
Thecatcher
Pythagorean
Theorem
home plate to second base.
b. The diagram below illustrates the throw made by
the third baseman.
So, the base runner is about 44 feet from first base to
second base. Therefore, the distance the base runner
is from second is about 90 – 44 or 46 feet.
Determine whether each set of measures can
be the lengths of the sides of a right triangle.
6. 8, 12, 16
SOLUTION: Since the measure of the longest side is 16, let c =
2
16, a = 8, and b = 12. Then determine whether c =
2
2
a +b .
The length of a is 90 – 15 or 75 feet. Find the
length of the throw c by using the Pythagorean
Theorem, substituting 75 for a and 90 for b.
2
2
2
No, because c ≠ a + b , a triangle with side lengths
8, 12, and 16 is not a right triangle.
7. 28, 45, 53
The third baseman has to throw the ball about 117
feet to the first baseman.
SOLUTION: Since the measure of the longest side is 53, let c =
2
53, a = 28, and b = 45. Then determine whether c =
2
2
a +b .
c. The diagram below illustrates the position of the
base runner.
2
2
2
Yes, because c = a + b , a triangle with side
lengths 28, 45, and 53 is a right triangle.
8. 7, 24, 25
Use the Pythagorean Theorem to find the length of
side a by substituting 90 for b and 100 for c.
SOLUTION: Since the measure of the longest side is 25, let c =
2
25, a = 7, and b = 24. Then determine whether c =
2
2
a +b .
2
So, the base runner is about 44 feet from first base to
second base. Therefore, the distance the base runner
is from second is about 90 – 44 or 46 feet.
eSolutions Manual - Powered by Cognero
Determine whether each set of measures can
be the lengths of the sides of a right triangle.
6. 8, 12, 16
2
2
Yes, because c = a + b , a triangle with side
lengths 7, 24, and 25 is a right triangle.
9. 15, 25, 45
SOLUTION: Page 2
Since the measure of the longest side is 45, let c =
2
45, a = 15, and b = 25. Then determine whether c =
2
2
2
10-5Yes,
The because
Pythagorean
c = a Theorem
+ b , a triangle with side
lengths 7, 24, and 25 is a right triangle.
9. 15, 25, 45
SOLUTION: Since the measure of the longest side is 45, let c =
2
45, a = 15, and b = 25. Then determine whether c =
2
12. SOLUTION: 2
a +b .
Use the Pythagorean Theorem, substituting
a and 20 for b.
2
2
for 2
No, because c ≠ a + b , a triangle with side lengths
15, 25, and 45 is not a right triangle.
Find each missing length. If necessary, round to
the nearest hundredth.
13. SOLUTION: Use the Pythagorean Theorem, substituting 9 for b
and 31 for c.
10. SOLUTION: Use the Pythagorean Theorem, substituting 6 for a
and 14 for b.
14. 11. SOLUTION: Use the Pythagorean Theorem, substituting 16 for a
and 26 for c.
SOLUTION: Use the Pythagorean Theorem, substituting 2 for a
and 12 for c.
15. SOLUTION: 12. eSolutions Manual - Powered by Cognero
Use the Pythagorean Theorem, substituting
SOLUTION: Use the Pythagorean Theorem, substituting
for and
for b.
Page 3
for a
10-5 The Pythagorean Theorem
15. 18. SOLUTION: Use the Pythagorean Theorem, substituting
and
for a
SOLUTION: Use the Pythagorean Theorem, substituting
for b.
and
16. SOLUTION: Use the Pythagorean Theorem, substituting 7 for b
and 25 for c.
17. SOLUTION: Use the Pythagorean Theorem, substituting 5 for a
and
for c.
for b
for c.
19. TELEVISION Larry is buying an entertainment
stand for his television. The diagonal of his television
is 42 inches. The space for the television measures
30 inches by 36 inches. Will Larry’s television fit?
Explain.
SOLUTION: Use the Pythagorean Theorem, substituting 30 for a
and 36 for b.
Yes; sample answer: The diagonal of the space in
the TV stand is about 46.9 inches, which is greater
than the diagonal of the television, so Larry’s TV will
fit.
Determine whether each set of measures can
be the lengths of the sides of a right triangle.
Then determine whether they form a
Pythagorean triple.
20. 9, 40, 41
SOLUTION: Since the measure of the longest side is 41, let c =
2
41, a = 9, and b = 40. Then determine whether c =
2
2
a +b .
18. eSolutions
Manual - Powered by Cognero
SOLUTION: Use the Pythagorean Theorem, substituting
Page 4
for b
3,
, and
is not a right triangle.
No, because a Pythagorean triple is a group of three
Yes; sample answer: The diagonal of the space in
the TV stand is about 46.9 inches, which is greater
10-5than
Thethe
Pythagorean
Theorem
diagonal of the
television, so Larry’s TV will
fit.
Determine whether each set of measures can
be the lengths of the sides of a right triangle.
Then determine whether they form a
Pythagorean triple.
20. 9, 40, 41
2
2
2
whole numbers that satisfy the equation c = a + b ,
where c is the greatest number.
22. SOLUTION: Since the measure of the longest side is 12, let c =
2
12, a = 4, and b =
. Then determine whether c
2
SOLUTION: Since the measure of the longest side is 41, let c =
2
=a +b .
2
41, a = 9, and b = 40. Then determine whether c =
2
2
a +b .
2
2
2
2
2
No, because c ≠ a + b , a triangle with side lengths
4,
, and 12 is not a right triangle.
No, because a Pythagorean triple is a group of three
2
Yes, because c = a + b , a triangle with side
lengths 9, 40, and 41 is a right triangle.
Yes, because a Pythagorean triple is a group of three
2
2
2
whole numbers that satisfy the equation c = a + b ,
where c is the greatest number.
2
2
2
whole numbers that satisfy the equation c = a + b ,
where c is the greatest number.
23. SOLUTION: Since the measure of the longest side is 14, let c =
2
14, a =
, and b = 7. Then determine whether c =
21. SOLUTION: 2
2
a +b .
Since the measure of the longest side is
, let c =
, a = 3, and b =
. Then determine whether
2
2
2
c =a +b .
2
2
2
No, because c ≠ a + b , a triangle with side lengths
, 7, and 14 is not a right triangle.
No, because a Pythagorean triple is a group of three
2
2
2
No, because c ≠ a + b , a triangle with side lengths
3,
, and
is not a right triangle.
No, because a Pythagorean triple is a group of three
2
2
2
whole numbers that satisfy the equation c = a + b ,
where c is the greatest number.
22. 2
2
2
24. 8, 31.5, 32.5
SOLUTION: Since the measure of the longest side is 32.5, let c =
2
32.5, a = 8, and b = 31.5. Then determine whether c
2
SOLUTION: Since the measure of the longest side is 12, let c =
2
12, a = 4, and b =
. Then determine whether c
2
whole numbers that satisfy the equation c = a + b ,
where c is the greatest number.
2
=a +b .
2
=a +b .
eSolutions Manual - Powered by Cognero
2
2
2
Page 5
Yes, because c = a + b , a triangle with side
lengths 8, 31.5, and 32.5 is a right triangle.
No, because a Pythagorean triple is a group of three
, 7, and 14 is not a right triangle.
No, because a Pythagorean triple is a group of three
2
2
2
10-5whole
The Pythagorean
Theorem
numbers that satisfy
the equation c = a + b ,
where c is the greatest number.
24. 8, 31.5, 32.5
26. 18, 24, 30
SOLUTION: Since the measure of the longest side is 32.5, let c =
2
32.5, a = 8, and b = 31.5. Then determine whether c
2
,
, and
is not a right triangle.
No, because a Pythagorean triple is a group of three
2
2
2
whole numbers that satisfy the equation c = a + b ,
where c is the greatest number.
2
=a +b .
2
2
2
Yes, because c = a + b , a triangle with side
lengths 8, 31.5, and 32.5 is a right triangle.
No, because a Pythagorean triple is a group of three
2
2
2
whole numbers that satisfy the equation c = a + b ,
where c is the greatest number, but b and c are not
whole numbers.
SOLUTION: Since the measure of the longest side is 30, let c =
2
30, a = 18, and b = 24. Then determine whether c =
2
2
a +b .
2
2
2
Yes, because c = a + b , a triangle with side
lengths 18, 24, and 30 is a right triangle.
Yes, because a Pythagorean triple is a group of three
2
2
2
whole numbers that satisfy the equation c = a + b ,
where c is the greatest number.
27. 36, 77, 85
25. SOLUTION: Since the measure of the longest side is 85, let c =
SOLUTION: 2
Since the measure of the longest side is
,a =
, and b =
2
2
, let c =
. Then determine
85, a = 36, and b = 77. Then determine whether c =
2
2
a +b .
2
whether c = a + b .
2
2
2
2
No, because c ≠ a + b , a triangle with side lengths
,
, and
is not a right triangle.
No, because a Pythagorean triple is a group of three
2
2
2
whole numbers that satisfy the equation c = a + b ,
where c is the greatest number.
26. 18, 24, 30
SOLUTION: Since the measure of the longest side is 30, let c =
2
30, a = 18, and b = 24. Then determine whether c =
2
2
a +b .
2
2
Yes, because c = a + b , a triangle with side
lengths 36, 77, and 85 is a right triangle.
Yes, because a Pythagorean triple is a group of three
2
2
2
whole numbers that satisfy the equation c = a + b ,
where c is the greatest number.
28. 17, 33, 98
SOLUTION: Since the measure of the longest side is 98, let c =
98, a = 17, and b = 33. The length of c is greater
than the sum of a and b, so this is not a triangle. 2
2
2
No, because c ≠ a + b , a triangle with side lengths
17, 33, and 98 is not a triangle.
No, because a Pythagorean triple is a group of three
2
2
2
whole numbers that satisfy the equation c = a + b ,
where c is the greatest number.
29. GEOMETRY Refer to the triangle shown.
eSolutions Manual - Powered by Cognero
2
2
2
Yes, because c = a + b , a triangle with side
lengths 18, 24, and 30 is a right triangle.
Page 6
No, because c ≠ a + b , a triangle with side lengths
17, 33, and 98 is not a triangle.
No, because a Pythagorean triple is a group of three
2
2
2
10-5whole
The Pythagorean
Theorem
numbers that satisfy
the equation c = a + b ,
where c is the greatest number.
29. GEOMETRY Refer to the triangle shown.
a. What is a?
b. Find the area of the triangle.
2
2
2
8 + 15 = 17 so the pieces form a right triangle by
the converse of the Pythagorean Theorem.
31. LADDER Mr. Takeo is locked out of his house.
The only open window is on the second floor. There
is a bush along the edge of the house, so he places
the neighbor’s ladder 10 feet from the house. To the
nearest foot, what length of ladder does he need to
reach the window?
SOLUTION: a. Use the Pythagorean Theorem, substituting 11 for
b and 23 for c.
SOLUTION: Use the Pythagorean Theorem, substituting 10 for a
and 28 for b.
b.
2
He will need a 30-ft ladder to reach the window.
The area of the triangle is about 111.1 units .
CCSS TOOLS Find the length of the
hypotenuse. Round to the nearest hundredth.
30. GARDENING Khaliah wants to plant flowers in a
triangular plot. She has three lengths of plastic
garden edging that measure 8 feet, 15 feet, and 17
feet. Determine whether these pieces form a right
triangle. Explain.
SOLUTION: Yes; sample answer: The longest side is 17 feet so
this has to be the hypotenuse.
32. SOLUTION: Use the Pythagorean Theorem, substituting 7 for a
and 10 for b.
2
2
2
8 + 15 = 17 so the pieces form a right triangle by
the converse of the Pythagorean Theorem.
31. LADDER Mr. Takeo is locked out of his house.
The only open window is on the second floor. There
is a bush along the edge of the house, so he places
the neighbor’s ladder 10 feet from the house. To the
nearest foot, what length of ladder does he need to
reach the window?
eSolutions Manual - Powered by Cognero
Page 7
10-5 The Pythagorean Theorem
The height of the roof is 10.6 in.
35. GEOMETRY Each side of a cube is 5 inches long.
Find the length of a diagonal of the cube.
SOLUTION: 33. SOLUTION: Use the Pythagorean Theorem, substituting 4 for a
and 7 for b.
34. DOLLHOUSE Alonso is building a dollhouse for
his sister’s birthday. The roof is 24 inches across and
the slanted side is 16 inches long as shown. Find the
height of the roof to the nearest tenth of an inch.
Use the Pythagorean Theorem to find the diagonal of
a side of the cube, substituting 5 for a and 5 for b.
Then, use the Pythagorean Theorem to find a
diagonal of the cube, substituting 5 for a and
for b.
SOLUTION: Use the Pythagorean Theorem, substituting 12 for b
and 16 for c.
The length of a diagonal of the cube is about 8.66 in.
The height of the roof is 10.6 in.
35. GEOMETRY Each side of a cube is 5 inches long.
Find the length of a diagonal of the cube.
SOLUTION: 36. TOWN SQUARES The largest town square in the
world is Tiananmen Square in Beijing, China,
covering 98 acres.
a. One square mile is 640 acres. Assuming that
Tiananmen Square is a square, how many feet long is
a side to the nearest foot?
b. To the nearest foot, what is the diagonal distance
across Tiananmen Square?
SOLUTION: a. First, use the conversion factor 1 square mile =
640 acres to convert the area of the square from
acres to square miles.
eSolutions Manual - Powered by Cognero
Use the Pythagorean Theorem to find the diagonal of
a side of the cube, substituting 5 for a and 5 for b.
Page 8
So, 98 acres is 0.153125 square miles. Use
10-5 The Pythagorean Theorem
The length of a diagonal of the cube is about 8.66 in.
36. TOWN SQUARES The largest town square in the
world is Tiananmen Square in Beijing, China,
covering 98 acres.
a. One square mile is 640 acres. Assuming that
Tiananmen Square is a square, how many feet long is
a side to the nearest foot?
b. To the nearest foot, what is the diagonal distance
across Tiananmen Square?
The diagonal distance across Tiananmen Square is
about 2922 feet.
37. TRUCKS Violeta needs to construct a ramp to roll
a cart of moving boxes from her garage into the back
of her truck. The truck is 6 feet from the garage.
The back of the truck is 36 inches above the ground.
How long does the ramp have to be?
SOLUTION: First, rewrite 36 inches as 3 ft. Use the Pythagorean
thereom, substituting 6 for a and 3 for b.
SOLUTION: a. First, use the conversion factor 1 square mile =
640 acres to convert the area of the square from
acres to square miles.
The ramp must be about 6.7 ft.
So, 98 acres is 0.153125 square miles. Use
dimensional analysis to convert this measure to
square feet.
If c is the measure of the hypotenuse of a right
triangle, find each missing measure. If
necessary, round to the nearest hundredth.
38. a = x, b = x + 41, c = 85
SOLUTION: Next, use the formula for the area of a square to find
the length of a side.
Use the Zero Product Property to solve for x. a = 36; b = 77
Therefore, a side of Tiananmen Square is about 2066
feet long.
b. Use the Pythagorean Theorem to find the length
of the diagonal of the square, substituting 2066 for
both a and b.
39. a = 8, b = x, c = x + 2
SOLUTION: b = 15; c = 17
40. a = 12, b = x − 2, c = x
The diagonal distance across Tiananmen Square is
about 2922 feet.
37. TRUCKS Violeta needs to construct a ramp to roll
a cart of moving boxes from her garage into the back
of her
truck.
The truck
is 6 feet from the garage.
eSolutions
Manual
- Powered
by Cognero
The back of the truck is 36 inches above the ground.
How long does the ramp have to be?
SOLUTION: Page 9
b = 35; c = 37
10-5 The Pythagorean Theorem
b = 15; c = 17
40. a = 12, b = x − 2, c = x
SOLUTION: Use the Zero Product Property to solve for x. a = 9; b = 40; c = 41
44. GEOMETRY A right triangle has one leg that is 8
inches shorter than the other leg. The hypotenuse is
30 inches long. Find the length of each leg.
SOLUTION: b = 35; c = 37
41. a = x, b = x + 7, c = 97
SOLUTION: Use the Zero Product Property to solve for x. a = 65; b = 72
42. a = x − 47, b = x, c = x + 2
SOLUTION: Use the Zero Product Property to solve for x. a = 16; b = 63; c = 65
43. a = x − 32, b = x − 1, c = x
SOLUTION: Use the Zero Product Property to solve for x. a = 9; b = 40; c = 41
44. GEOMETRY A right triangle has one leg that is 8
inches shorter than the other leg. The hypotenuse is
30 inches long. Find the length of each leg.
SOLUTION: eSolutions Manual - Powered by Cognero
The length of each leg of the triangle is about 24.83
in. and about 16.83 in.
45. MULTIPLE REPRESENTATIONS In this
problem, you will derive a method for finding the
midpoint and length of a segment on the coordinate
plane.
a. Graphical Use a graph to find the lengths of the
segments between (3, 2) and (8, 2) and between (4,
1) and (4, 9). Then find the midpoint of each
segment.
b. Logical Use what you learned in part a to write
expressions for the lengths of the segments between
(x1 , y) and (x2 , y) and between (x, y 1 ) and (x, y 2).
What would be the midpoint of each segment?
c. Analytical Based on your results from part b, find
the midpoint of the segment with endpoints at (x1,
y 1), and (x2, y 2).
d. Analytical Use the Pythagorean Theorem to
write an expression for the distance between (x1, y 1),
and (x2, y 2).
SOLUTION: a. From the graph, the length of the vertical segment
is 9 – 1 or 8 units and the length of the horizontal line
is 8 – 3 or 5 units.
Page 10
and (x2, y 2).
SOLUTION: From
the graph, the
length of the vertical segment
10-5a. The
Pythagorean
Theorem
is 9 – 1 or 8 units and the length of the horizontal line
is 8 – 3 or 5 units.
2
2
2
We know the Pythagorean Theorem is a + b = c .
We can solve this formula for c by taking the square
root of each side. So,
. Now we
can replace the variables with the lengths of each
side. The length of side a is x2 – x1. The length of
side b is y 2 – y 1. The length of side c is now
.
The midpoints will be located in the middle of each
segment, or the average of the coordinates. For the
vertical segment, the value between 1 and 9 is 5, so
the midpoint should be at (4, 5). For the horizontal
segment, the value between 3 and 8 is 5.5, so the
midpoint should be at (5.5, 2).
b. Distance: The length of the segment between (x1,
y) and (x2, y) will be the absolute value of x1 – x2.
46. ERROR ANALYSIS Wyatt and Dario are
determining whether 36, 77, and 85 form a
Pythagorean triple. Is either of them correct? Explain
your reasoning.
The length of the segment between (x, y 1) and (x,
y2) will be the absolute value of y 1 – y 2.
Midpoint: When the x-coordinates are identical, then
the x-coordinate of the midpoint will also be the same
value. This is true for the y-coordinates as well. The
midpoint of (x1, y) and (x2, y) is at the average of the
x-coordinates and y or at
, and the
midpoint of (x, y 1) and (x, y 2) is at
.
c. When the x- and y-coordinates are both different,
the midpoint should be at the average values of the
.
coordinates, or at
d. Form a right triangle with the segment formed by
the two points as the hypotenuse. Note that the other
coordinate is (x2, y 1) in the drawing. 2
2
2
We know the Pythagorean Theorem is a + b = c .
eSolutions
Manual
- Powered
by Cognero
We can
solve
this formula
for c by taking the square
root of each side. So,
. Now we
can replace the variables with the lengths of each
SOLUTION: Wyatt is correct. The square of the greatest value
should be equal to the sum of the squares of the two
smaller values. Since this is the case, the numbers form a Pythagorean triple.
47. CCSS PERSEVERANCE Find the value of x in
the figure shown.
SOLUTION: The figure can be separated into two right triangles.
Let m represent the missing side length common to
both right triangles. From the Pythagorean Theorem,
2
2
2
2
2
2
m = 2 + x and 14 = 8 + m . Using substitution,
2
2
2
2
you can find that 2 + x = 14 – 8 .
Solve for x.
Page 11
SOLUTION: Wyatt is correct. The square of the greatest value
should be equal to the sum of the squares of the two
10-5smaller values. Since this is the case, the numbers The Pythagorean Theorem
form a Pythagorean triple.
47. CCSS PERSEVERANCE Find the value of x in
the figure shown.
area of 6 cm . A right triangle with legs measuring 2
cm and
cm also has a hypotenuse of 5 cm, but its area is
2
cm , which is not equivalent to 6
2
cm .
49. OPEN ENDED Draw a right triangle that has a
hypotenuse of
units.
SOLUTION: Use the Pythagorean Theorem to select two side
lengths for a and b with the given value of c.
SOLUTION: The figure can be separated into two right triangles.
Let m represent the missing side length common to
both right triangles. From the Pythagorean Theorem,
2
2
2
2
2
2
m = 2 + x and 14 = 8 + m . Using substitution,
2
2
2
2
you can find that 2 + x = 14 – 8 .
Solve for x.
50. WRITING IN MATH Explain how to determine
whether segments in three lengths could form a right
triangle.
48. REASONING Provide a counterexample to the
statement.
Any two right triangles with the same hypotenuse
have the same area.
SOLUTION: From the converse of the Pythagorean Theorem, if
2
2
2
a + b = c then a, b, and c are the lengths of the
side of a right triangle. So, check to see whether the
square of the greatest number is equal to the sum of
the squares of the other two numbers.
Consider triangle with sides of 9, 12 and 15 and a
triangle with sides of 7, 13 and 19. SOLUTION: Different lengths of a and b can produce the same
value of c in the Pythagorean theorem.
51. GEOMETRY Find the missing length.
Sample answer: A right triangle with legs measuring
3 cm and 4 cm has a hypotenuse of 5 cm and an
2
area of 6 cm . A right triangle with legs measuring 2
cm and
cm also has a hypotenuse of 5 cm, but its area is
2
cm , which is not equivalent to 6
2
cm .
A −17
B −
C D 17
SOLUTION: 49. OPEN ENDED Draw a right triangle that has a
hypotenuse of
units.
eSolutions Manual - Powered by Cognero
SOLUTION: Use the Pythagorean Theorem to select two side
Page 12
10-5 The Pythagorean Theorem
The correct choice is H.
51. GEOMETRY Find the missing length.
53. SHORT RESPONSE A plumber charges $40 for
the first hour of each house call plus $8 for each
additional half hour. If the plumber works for 4 hours,
how much does he charge?
SOLUTION: Let x = the total number of hours worked.
A −17
B −
C D 17
SOLUTION: The plumber charges $88 for 4 hours of work.
54. Find the next term in the geometric
sequence
.
A B
The correct choice is C.
52. What is a solution of this equation?
C D F 0, 3
G 3
H 0
J no solutions
SOLUTION: SOLUTION: The correct choice is B.
The correct choice is H.
Solve each equation. Check your solution.
53. SHORT RESPONSE A plumber charges $40 for
the first hour of each house call plus $8 for each
additional half hour. If the plumber works for 4 hours,
how much does he charge?
55. SOLUTION: SOLUTION: Let x = the total number of hours worked.
Check.
eSolutions
- Powered
by $88
Cognero
The Manual
plumber
charges
for
4 hours of work.
54. Find the next term in the geometric
Page 13
56. 10-5 The Pythagorean Theorem
The correct choice is B.
Solve each equation. Check your solution.
58. 55. SOLUTION: SOLUTION: Check.
Check.
56. SOLUTION: 59. SOLUTION: Check.
Check.
57. SOLUTION: 60. SOLUTION: Check.
Check.
58. SOLUTION: There is no real solution.
Simplify each expression.
eSolutions Manual - Powered by Cognero
61. Check.
SOLUTION: Page 14
10-5 The Pythagorean Theorem
There is no real solution.
Simplify each expression.
61. Describe how the graph of each function is
2
related to the graph of f (x) = x .
2
SOLUTION: 67. g(x) = x − 8
SOLUTION: 2
The graph of f (x) = x + c represents a vertical
translation of the parent graph. The value of c is –8,
2
and –8 < 0. If c < 0, the graph of f (x) = x is
translated units down. Therefore, the graph of g
62. SOLUTION: 2
(x) = x – 8 is a translation of the parent graph
shifted down 8 units.
68. h(x) =
63. SOLUTION: x
2
SOLUTION: 2
The graph of f (x) = ax stretches or compresses the
2
graph of f (x) = x vertically. The value of a is
< 1. If 0 < and 0 <
,
< 1, the graph of f (x) = x
2
is compressed vertically. Therefore, the graph of h
64. (x) =
SOLUTION: 2
x is the parent graph compressed vertically.
2
69. h(x) = −x + 5
SOLUTION: 2
The graph of f (x) = –x reflects the graph of f (x) =
2
2
x across the x-axis. The graph of f (x) = x + c
represents a vertical translation of the parent graph.
The value of c is 5, and 5 > 0. If c > 0, the graph of f
2
(x) = x is translated units up. Therefore, the 65. SOLUTION: 2
graph of h(x) = −x + 5 is a translation of the parent
graph shifted up 8 units and reflected across the xaxis.
2
70. g(x) = (x + 10)
66. SOLUTION: SOLUTION: 2
The graph of f (x) = (x – c) represents a horizontal
translation of the parent graph. The value of c is –10,
2
and –10 < 0. If c < 0, the graph of f (x) = (x – c) is
translated units left. Therefore, the graph of g(x)
2
Describe how the graph of each function is
2
related to the graph of f (x) = x .
= (x + 10) is a translation of the parent graph
shifted left 10 units.
eSolutions Manual - Powered by Cognero
2
67. g(x) = x − 8
71. g(x) = −2x
2
SOLUTION: Page 15
2
Therefore, the graph of h(x) = −x
units up. Therefore, the (x) = x is translated
2
is a
2
graph of h(x) = −x + 5 is a translation of the parent
shifted up 8 units
and reflected across the x10-5graph
The Pythagorean
Theorem
axis.
2
70. g(x) = (x + 10)
SOLUTION: 2
The graph of f (x) = (x – c) represents a horizontal
translation of the parent graph. The value of c is –10,
2
and –10 < 0. If c < 0, the graph of f (x) = (x – c) is
translated units left. Therefore, the graph of g(x)
units and reflected across the x-axis.
73. ROCK CLIMBING While rock climbing, Damaris
launches a grappling hook from a height of 6 feet
with an initial upward velocity of 56 feet per second.
The hook just misses the stone ledge that she wants
to scale. As it falls, the hook anchors on a ledge 30
feet above the ground. How long was the hook in the
air?
SOLUTION: 2
= (x + 10) is a translation of the parent graph
shifted left 10 units.
71. g(x) = −2x
translation of the parent graph shifted down
2
SOLUTION: 2
The graph of f (x) = –x reflects the graph of f (x) =
2
2
x across the x-axis. The graph of f (x) = ax
2
stretches or compresses the graph of f (x) = x
vertically. The value of a is –2, and |–2| > 1. If
> 2
1, the graph of f (x) = x is stretched vertically.
2
Therefore, the graph of g(x) = –2x is the parent
graph reflected across the x-axis and stretched
vertically.
A half second after she throws the hook, it is 30 feet
in the air. The hook continues up, then drops back
down to the height of 30 feet 3 seconds after the
throw. The hook was in the air for 3 seconds.
Find each product.
74. (b + 8)(b + 2)
SOLUTION: 2
72. h(x) = −x −
SOLUTION: 2
The graph of f (x) = –x reflects the graph of f (x) =
2
2
x across the x-axis. The graph of f (x) = x + c
represents a vertical translation of the parent graph.
The value of c is
, and
SOLUTION: < 0. If c < 0, the
2
graph of f (x) = x is translated
units down. Therefore, the graph of h(x) = −x
2
76. (y + 4)(y − 8)
SOLUTION: is a
translation of the parent graph shifted down
units and reflected across the x-axis.
73. ROCK CLIMBING While rock climbing, Damaris
launches a grappling hook from a height of 6 feet
with an initial upward velocity of 56 feet per second.
The hook just misses the stone ledge that she wants
to scale. As it falls, the hook anchors on a ledge 30
feet above the ground. How long was the hook in the
air?
SOLUTION: 75. (x − 4)(x − 9)
77. (p + 2)(p − 10)
SOLUTION: 78. (2w − 5)(w + 7)
SOLUTION: eSolutions Manual - Powered by Cognero
Page 16
79. (8d + 3)(5d + 2)
77. (p + 2)(p − 10)
SOLUTION: 10-5 The Pythagorean Theorem
78. (2w − 5)(w + 7)
82. SOLUTION: SOLUTION: 79. (8d + 3)(5d + 2)
SOLUTION: 83. 80. BUSINESS The amount of money spent at West
Outlet Mall continues to increase. The total T(x) in
millions of dollars can be estimated by the function T
SOLUTION: x
(x) = 12(1.12) , where x is the number of years after
it opened in 2005. Find the amount of sales in 2015,
2016, and 2017.
SOLUTION: 84. SOLUTION: The sales for the mall in 2015 will be about $37.27
million; in 2016 will be about $41.74 million; and in
2017 will be about $46.75 million.
Solve each proportion.
81. SOLUTION: 82. SOLUTION: eSolutions Manual - Powered by Cognero
83. Page 17
10-6 Trigonometric Ratios
Find the values of the three trigonometric
ratios for angle A .
3. 1. SOLUTION: SOLUTION: 4. SOLUTION: 2. SOLUTION: CCSS TOOLS Use a calculator to find the value
of each trigonometric ratio to the nearest tenthousandth.
5. sin 37°
SOLUTION: 3. SOLUTION: eSolutions Manual - Powered by Cognero
Keystrokes:
37
Then, sin 37° = 0.6018. 6. cos 23°
SOLUTION: Page 1
SOLUTION: : Ratios
37
10-6Keystrokes
Trigonometric
Then, sin 37° = 0.6018. 6. cos 23°
SOLUTION: 10. Keystrokes:
23
Then, cos 23° = 0.9205
SOLUTION: Find the measure of
7. tan 14°
.
SOLUTION: Keystrokes:
14
Then, tan 14° = 0.2493.
Find
.
8. cos 82°
SOLUTION: Keystrokes:
82
Then, cos 82° = 0.1392.
Find
.
Solve each right triangle. Round each side
length to the nearest tenth.
9. SOLUTION: Find the measure of
.
SOLUTION: Find the measure of
Find
11. .
.
Find
.
Find
.
Find
.
10. SOLUTION: Find the measure of
eSolutions Manual - Powered by Cognero
.
Page 2
10-6 Trigonometric Ratios
The length of the run of the hill is about 11,326.2 ft.
Find m X for each right triangle to the nearest
degree.
12. SOLUTION: Find the measure of
14. .
SOLUTION: –1
Use tan
Find
on a calculator.
Keystrokes:
.
-1
[TAN ]
Find
.
15. SOLUTION: 13. SNOWBOARDING A hill used for snowboarding
has a vertical drop of 3500 feet. The angle the run
makes with the ground is 18°. Estimate the length of r.
–1
Use cos
on a calculator.
Keystrokes:
-1
[COS ]
SOLUTION: The length of the run of the hill is about 11,326.2 ft.
Find m X for each right triangle to the nearest
degree.
16. SOLUTION: –1
Use tan
on a calculator.
Keystrokes:
-1
[TAN ]
14. SOLUTION: –1
Use Manual
tan on
a calculator.
eSolutions
- Powered
by Cognero
Keystrokes:
-1
[TAN ]
Page 3
Keystrokes:
Keystrokes:
[TAN ]
[SIN ]
10-6 Trigonometric Ratios
Find the values of the three trigonometric
ratios for angle B.
17. SOLUTION: Use sin
–1
18. on a calculator.
Keystrokes:
SOLUTION: Find
-1
[SIN ]
.
Find the values of the three trigonometric
ratios for angle B.
18. SOLUTION: Find
.
19. SOLUTION: Find
.
eSolutions Manual - Powered by Cognero
Page 4
10-6 Trigonometric Ratios
19. 20. SOLUTION: Find
.
SOLUTION: Find
.
CCSS TOOLS Use a calculator to find the value
of each trigonometric ratio to the nearest tenthousandth.
21. tan 2°
20. SOLUTION: SOLUTION: Find
Keystrokes:
2
Then, tan 2° = 0.0349.
.
22. sin 89°
SOLUTION: Keystrokes:
89
Then, sin 89° = 0.9998.
eSolutions Manual - Powered by Cognero
Page 5
23. cos 44°
SOLUTION: 28. tan 60°
SOLUTION: SOLUTION: Keystrokes:
2
10-6Then, tan 2° = 0.0349.
Trigonometric Ratios
Keystrokes:
60
Then, tan 60° = 1.7321.
22. sin 89°
Solve each right triangle. Round each side
length to the nearest tenth.
SOLUTION: Keystrokes:
89
Then, sin 89° = 0.9998.
23. cos 44°
29. SOLUTION: SOLUTION: Find the measure of
Keystrokes:
44
Then, cos 44° = 0.7193.
24. tan 45°
Find
SOLUTION: Keystrokes:
Then, tan 45° = 1.
.
.
45
25. sin 73°
SOLUTION: Keystrokes:
73
Then, sin 73° = 0.9563
Find
.
26. cos 90°
SOLUTION: Keystrokes:
Then, cos 90° = 0.
90
27. sin 30°
30. SOLUTION: SOLUTION: Find the measure of
Keystrokes:
30
Then, sin 30° = 0.5.
.
28. tan 60°
SOLUTION: Find
.
Keystrokes:
60
Then, tan 60° = 1.7321.
Solve each right triangle. Round each side
length to the nearest tenth.
Find
.
29. SOLUTION: Find the measure of
.
eSolutions Manual - Powered by Cognero
Page 6
31. 10-6 Trigonometric Ratios
31. SOLUTION: Find the measure of
.
33. SOLUTION: Find the measure of
Find
.
.
Find
.
Find
.
Find
.
32. SOLUTION: Find the measure of
.
34. SOLUTION: Find the measure of
.
Find
.
Find
.
Find
.
Find
eSolutions Manual - Powered by Cognero
.
Page 7
35. ESCALATORS At a local mall, an escalator is 110
feet long. The angle the escalator makes with the
10-6 Trigonometric Ratios
The escalator is about 53 ft high.
Find m J for each right triangle to the nearest
degree.
34. SOLUTION: Find the measure of
36. .
SOLUTION: You know the measure of the side opposite ∠J and
the measure of the hypotenuse. Use the sine ratio.
Find
.
–1
Use a calculator and the [sin ]function to find the
measure of the angle.
Keystrokes:
Find
.
-1
[SIN ] 10
24
24.62431835
So, m∠J ≈ 25°.
35. ESCALATORS At a local mall, an escalator is 110
feet long. The angle the escalator makes with the
ground is 29°. Find the height reached by the escalator.
37. SOLUTION: SOLUTION: You know the measure of the side opposite ∠J and
the measure of the hypotenuse. Use the sine ratio.
–1
Use a calculator and the [sin ] function to find the
measure of the angle.
The escalator is about 53 ft high.
Find m J for each right triangle to the nearest
degree.
Keystrokes:
-1
[SIN ] 15
17
61.92751306
So, m∠J ≈ 62°.
36. SOLUTION: eSolutions
Powered
by Cognero
YouManual
know -the
measure
of the side opposite ∠J and
the measure of the hypotenuse. Use the sine ratio.
Page 8
61.92751306
So, m∠J ≈ 62°.
10-6 Trigonometric Ratios
30.96375653
So, m∠J ≈ 31°.
40. SOLUTION: You know the measure of the side adjacent to ∠J
and the measure of the hypotenuse. Use the cosine
ratio.
38. SOLUTION: You know the measure of the side opposite ∠J and
the measure of the side adjacent to ∠J. Use the
tangent ratio.
–1
Use a calculator and the [cos ] function to find the
measure of the angle.
–1
Use a calculator and the [tan ] function to find the
measure of the angle.
Keystrokes:
Keystrokes:
-1
[TAN ] 23
-1
[COS ] 5
16
71.79004314
14
So, m∠J ≈ 72°.
58.67130713
So, m∠J ≈ 59°.
41. 39. SOLUTION: You know the measure of the side opposite ∠J and
the measure of the side adjacent to ∠J. Use the
tangent ratio.
SOLUTION: You know the measure of the side adjacent to ∠J
and the measure of the hypotenuse. Use the cosine
ratio.
–1
Use a calculator and the [tan ] function to find the
measure of the angle.
–1
Use a calculator and the [cos ] function to find the
measure of the angle.
Keystrokes:
-1
[TAN ] 6
10
30.96375653
Keystrokes:
-1
[COS ] 11
So, m∠J ≈ 31°.
49.67978493
So, m∠J ≈ 50°.
40. SOLUTION: eSolutions
Manual - Powered by Cognero
You know the measure of the side adjacent to ∠J
and the measure of the hypotenuse. Use the cosine
17 42. MONUMENTS The Lincoln Memorial building
measures 204 feet long, 134 feet wide, and 99 feet
tall. Chloe is looking at the top of the monument at an
angle of 55°. How far away is she standing from the monument?
Page 9
SOLUTION: 49.67978493
m∠J ≈ 50°. Ratios
10-6So,
Trigonometric
42. MONUMENTS The Lincoln Memorial building
measures 204 feet long, 134 feet wide, and 99 feet
tall. Chloe is looking at the top of the monument at an
angle of 55°. How far away is she standing from the monument?
SOLUTION: The horizontal distance to the city is about 35,577 ft.
44. FORESTS A forest ranger estimates the height of
a tree is about 175 feet. If the forest ranger is
standing 100 feet from the base of the tree, what is
the measure of the angle formed between the range
and the top of the tree?
SOLUTION: The angle formed between the ground and the top of
the tree is about 60°.
Chloe is standing about 69 ft away from the
monument.
43. AIRPLANES Ella looks down at a city from an
airplane window. The airplane is 5000 feet in the air,
and she looks down at an angle of 8°. Determine the horizontal distance to the city.
Suppose
ABC .
A is an acute angle of right triangle
45. Find sin A and tan A if cos
.
SOLUTION: SOLUTION: The horizontal distance to the city is about 35,577 ft.
44. FORESTS A forest ranger estimates the height of
a tree is about 175 feet. If the forest ranger is
standing 100 feet from the base of the tree, what is
the measure of the angle formed between the range
and the top of the tree?
eSolutions Manual - Powered by Cognero
SOLUTION: 46. Find tan A and cos A if sin
SOLUTION: .
Page 10
10-6 Trigonometric Ratios
46. Find tan A and cos A if sin
.
.
SOLUTION: SOLUTION: 47. Find cos A and tan A if sin
48. Find sin A and cos A if tan
.
SOLUTION: 49. SUBMARINES A submarine descends into the
ocean at an angle of 10° below the water line and travels 3 miles diagonally. How far beneath the
surface of the water has the submarine reached?
SOLUTION: The submarine went about 0.5 mi beneath the
surface of the water.
48. Find sin A and cos A if tan
.
50. MULTIPLE REPRESENTATIONS In this
problem, you will explore a relationship between the
sine and cosine functions.
SOLUTION: a. TABULAR Copy and complete the table using
the triangles shown above.
eSolutions Manual - Powered by Cognero
Page 11
b. VERBAL Make a conjecture about the sum of
10-6 Trigonometric Ratios
a. TABULAR Copy and complete the table using
the triangles shown above.
b. The sum of the squares of the sine and cosine of
an acute angle in a right triangle is equal to 1.
51. CHALLENGE Find a and c in the triangle shown.
b. VERBAL Make a conjecture about the sum of
the squares of the sine and cosine of an acute angle
of an acute angle in a right triangle.
SOLUTION: a.
SOLUTION: Use the sum of the angles of a triangle to determine
the value of a.
Use the value of a to determine the value of the
angles of the triangle.
The angles are 90°, 6(5) – 3 or 27°, and 12(5) + 3 or 63°.
Use a trigonometric ratio to find the value of c.
Therefore, a = 5 and c ≈ 7.3.
Then ,
52. REASONING Use the definitions of the sine and
cosine ratios to define the tangent ratio.
SOLUTION: Sin is defined as
and Cos is defined as
Tan can be defined as
.
because:
b. The sum of the squares of the sine and cosine of
an acute angle in a right triangle is equal to 1.
51. CHALLENGE Find a and c in the triangle shown.
eSolutions Manual - Powered by Cognero
Page 12
Therefore, a = 5 and c ≈ 7.3.
10-6 Trigonometric Ratios
52. REASONING Use the definitions of the sine and
cosine ratios to define the tangent ratio.
SOLUTION: Sin is defined as
and Cos is defined as
Tan can be defined as
.
54. CCSS ARGUMENTS The sine and cosine of an
acute angle in a right triangle are equal. What can
you conclude about the triangle?
SOLUTION: Given: ΔABC with sides a, b, and c as shown;
sin A = cos A
because:
53. WRITING IN MATH How can triangles be used
to solve problems?
SOLUTION: Many real world problems involve trying to determine
the correct height or length of a given structure.
When lengths and angles are known, right triangles
can be drawn, and trigonometric ratios can be used
to determine missing sides and angles. Similarly, other situations may require triangles and
the Pythagorean theorem to determine unknown
lengths. If a = b, then
. A triangle that has two
congruent sides is called an isosceles triangle.
Therefore, this triangle is an isosceles right triangle.
The legs of the right triangle are equal to each other.
55. WRITING IN MATH Explain how to use
trigonometric ratios to find the missing length of a
side of a right triangle given the measure of one
acute angle and the length of one side.
SOLUTION: Use the angle given and the measure of the known
side to set up one of the trigonometric ratios. The
sine ratio uses the opposite side and hypotenuse of
the triangle. The cosine ratio uses the adjacent side
and hypotenuse of the triangle. The tangent ratio
uses the opposite and adjacent sides of the triangle.
Choose the ratio that can be used to solve for the
unknown measure.
Given the following triangle find the missing sides a
and c.
54. CCSS ARGUMENTS The sine and cosine of an
acute angle in a right triangle are equal. What can
you conclude about the triangle?
eSolutions
Manual - Powered by Cognero
SOLUTION: Given: ΔABC with sides a, b, and c as shown;
sin A = cos A
Page 13
Since you know the measure of ∠A, set up the
uses the opposite and adjacent sides of the triangle.
Choose the ratio that can be used to solve for the
unknown measure.
10-6 Trigonometric Ratios
Given the following triangle find the missing sides a
and c.
56. Which graph below represents the solution set for −2
≤ x ≤ 4?
A
B
C
Since you know the measure of ∠A, set up the
trigonometric ratios for the acute angle of 42°.
Let a be the measure of the side opposite ∠A, 15 is
the measure of the side adjacent ∠A, and c is the
measure of the hypotenuse.
So, if you are trying to find the measure of a, use
the tangent ratio. If you are trying to find the
measure of c, use the cosine ratio.
D
SOLUTION: The inequality uses less than or equal to signs, so the
points on the graph must be solid. So, choices B and
D are incorrect. In the inequality, x is found between
the two values, so choice C in incorrect.
The correct choice is A.
57. PROBABILITY Suppose one chip is chosen from a bin with the chips shown. To the nearest tenth,
what is the probability that a green chip is chosen?
F 0.2
G 0.5
H 0.6
J 0.8
56. Which graph below represents the solution set for −2
≤ x ≤ 4?
A
SOLUTION: B
C
The correct choice is F.
58. In the graph, for what value(s) of x is y = 0?
D
SOLUTION: eSolutions
Manual - Powered by Cognero
The inequality uses less than or equal to signs, so the
points on the graph must be solid. So, choices B and
Page 14
10-6 Trigonometric Ratios
The correct choice is F.
a. Let h represent the height reached by the ladder.
Use the Pythagorean Theorem to represent the
value of h in terms of the other two sides.
58. In the graph, for what value(s) of x is y = 0?
If the bottom of the ladder is moved closer to the
base of the house, the distance the bottom of the
2
A 0
B −1
C 1
D 1 and −1
SOLUTION: The graph crosses the x-axis twice, so there are two
values of x for which y = 0. Choices A, B, and C
only offer one x value.
The correct choice is D.
59. EXTENDED RESPONSE A 16-foot ladder is
placed against the side of a house so that the bottom
of the ladder is 8 feet from the base of the house.
a. If the bottom of the ladder is moved closer to the
base of the house, does the height reached by the
ladder increase or decrease?
b. What conclusion can you make about the distance
between the bottom of the ladder and the base of the
house and the height reached by the ladder?
c. How high does the ladder reach if the ladder is 3
feet from the base of the house?
ladder is from the wall will decrease. When 16 is
2
subtracted by a number smaller than 8 , the
2
2
difference is greater than 16 - 8 . Since you are
finding the square root of a larger number, h will be
greater. Therefore, as the bottom of the ladder is
moved closer to the base of the house, the height
reached by the ladder will increase.
b. Sample answer: Let h represent the height
reached by the ladder and d represent the distance
between the bottom of the ladder and the base of the
house. The house is built perpendicular to the ground,
so the ladder will form a right triangle when it is
placed against the side of the house. Use the
Pythagorean Theorem to relate the sides of the
triangle.
2
Therefore, the sum of their squares is 16 or 256.
c.
SOLUTION: If the ladder is 3 ft from the base of the house, then it
reaches a height of about 15.7 ft.
If c is the measure of the hypotenuse of a right
triangle, find each missing measure. If
necessary, round to the nearest hundredth.
60. a = 16, b = 63, c = ?
SOLUTION: a. Let h represent the height reached by the ladder.
Use the Pythagorean Theorem to represent the
value of h in terms of the other two sides.
eSolutions Manual - Powered by Cognero
Page 15
The length of the hypotenuse is 65 units.
If the bottom of the ladder is moved closer to the
The length of one of the legs is
units.
the ladder is 3 ftRatios
from the base of the house, then it
10-6IfTrigonometric
reaches a height of about 15.7 ft.
If c is the measure of the hypotenuse of a right
triangle, find each missing measure. If
necessary, round to the nearest hundredth.
60. a = 16, b = 63, c = ?
or about 10.72
63. a = 6, b = 3, c = ?
SOLUTION: SOLUTION: The length of the hypotenuse is
units.
The length of the hypotenuse is 65 units.
64. 61. b = 3,
,c=?
or about 6.71 , c = 12, a = ?
SOLUTION: SOLUTION: The length of the hypotenuse is 11 units.
The length of one of the legs is
units.
or about 8.19 62. c = 14, a = 9, b = ?
65. a = 4,
SOLUTION: The length of one of the legs is
units.
63. a = 6, b = 3, c = ?
SOLUTION: ,c=?
SOLUTION: or about 10.72
The length of the hypotenuse is
units.
or about 5.20 66. AVIATION The relationship between a plane’s
length L in feet and the pounds P its wings can lift is
eSolutions Manual - Powered by Cognero
described by
, where k is the constant of
proportionality. A Boeing 747 is 232 feet long and
has a takeoff weight of 870,000 pounds. Determine k
Page 16
for this plane to the nearest hundredth.
SOLUTION: length of the hypotenuse
is
10-6The
Trigonometric
Ratios
units.
or about 5.20 66. AVIATION The relationship between a plane’s
length L in feet and the pounds P its wings can lift is
69. described by
, where k is the constant of
proportionality. A Boeing 747 is 232 feet long and
has a takeoff weight of 870,000 pounds. Determine k
for this plane to the nearest hundredth.
SOLUTION: SOLUTION: 70. The constant of proportionality is about 0.06.
SOLUTION: 67. FINANCIAL LITERACY A salesperson is paid
$32,000 a year plus 5% of the amount in sales made.
What is the amount of sales needed to have an
annual income greater than $45,000?
SOLUTION: Let x represent the amount of sales made.
71. The amount of sales must be more than $260,000.
Solve each proportion.
SOLUTION: 68. SOLUTION: 69. SOLUTION: eSolutions
Manual - Powered by Cognero
Page 17
. Another way to identify the stretch is to
notice that the y-values in the table are 2 times the
corresponding y-values for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ 0}.
Mid-Chapter Quiz
Graph each function. Compare to the parent
graph. State the domain and range.
2. SOLUTION: Make a table.
x
0
0.5
1
2
3
y
0
≈ –
–4
≈ –
≈ –
2.8
5.7
6.9
Plot the points and draw a smooth curve. 1. SOLUTION: Make a table.
x
0
0.5
1
2
3
y
0
≈ 1.4
2
≈ 2.8 ≈ 3.5
Plot the points and draw a smooth curve. 4
4
4
–8
The parent function
is multiplied by a value
less than 1, so the graph is a vertical stretch of
and a reflection across the x-axis. Another
way to identify the stretch is to notice that the yvalues in the table are –4 times the corresponding yvalues for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≤ 0}.
is multiplied by a value
The parent function greater than 1, so the graph is a vertical stretch of
. Another way to identify the stretch is to
notice that the y-values in the table are 2 times the
corresponding y-values for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ 0}.
3. 2. SOLUTION: Make a table.
x
0
0.5
1
2
3
y
0
≈ –
–4
≈ –
≈ –
2.8
5.7
6.9
Plot the points and draw a smooth curve. 4
–8
The parent function
is multiplied by a value
less than 1, so the graph is a vertical stretch of
Another
and a reflection across the x-axis.
eSolutions Manual
- Powered by Cognero
way to identify the stretch is to notice that the yvalues in the table are –4 times the corresponding y-
SOLUTION: Make a table.
x
0
0.5
1
2
3
y
0
0.5
≈ 0.4
≈ 0.7 ≈ 0.9
Plot the points and draw a smooth curve. 4
1
The parent function
is multiplied by a value
less than 1 and greater than 0, so the the graph is a
vertical compression of
. Another way to
Page 1
identify the compression is to notice that the y-values
in the table are
times the corresponding y-values
and a reflection across the x-axis. Another
way to identify the stretch is to notice that the yvalues in the table are –4 times the corresponding yvalues for Quiz
the parent function. The domain is {x|x ≥ Mid-Chapter
0} and the range is {y|y ≤ 0}.
identify the compression is to notice that the y-values
in the table are times the corresponding y-values
for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ 0}.
4. 3. SOLUTION: Make a table.
x
0
0.5
1
2
3
y
0
0.5
≈ 0.4
≈ 0.7 ≈ 0.9
Plot the points and draw a smooth curve. SOLUTION: Make a table.
x
0
0.5
1
2
3
y
–3
≈ –
–2
≈ –
≈ –
2.3
1.6
1.3
Plot the points and draw a smooth curve. 4
1
4
–1
The parent function
is multiplied by a value
less than 1 and greater than 0, so the the graph is a
vertical compression of
. Another way to
identify the compression is to notice that the y-values
The value 3 is being subtracted from the parent
function
, so the graph is translated down 3
units from the parent graph
. Another way to
identify the translation is to note that the y-values in
the table are 3 less than the corresponding y-values
for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ –3}.
in the table are times the corresponding y-values
for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ 0}.
5. 4. SOLUTION: Make a table.
x
0
0.5
1
2
3
y
–3
≈ –
–2
≈ –
≈ –
2.3
1.6
1.3
Plot the points and draw a smooth curve. 4
–1
SOLUTION: Make a table
x
1
1.5
2
3
y
0
1
≈ 0.7
≈ 1.4
Plot the points and draw a smooth curve. 4
≈ 1.7
The value 3 is being subtracted from the parent
function
, so the graph is translated down 3
eSolutions
Powered
bygraph
Cognero
unitsManual
from -the
parent
. Another way to
identify the translation is to note that the y-values in
the table are 3 less than the corresponding y-values
The value 1 is being subtracted from the square root
of the parent function
, so the graph is
translated 1 unit right from the parent graph
.
Another way to identify the translation is to notePage
that 2
the x-values in the table are 1 more than the
corresponding x-values for the parent function. The
units from the parent graph
. Another way to
identify the translation is to note that the y-values in
the table are 3 less than the corresponding y-values
for the parent
function. The domain is {x|x ≥ 0} and Mid-Chapter
Quiz
the range is {y|y ≥ –3}.
5. translated 1 unit right from the parent graph
.
Another way to identify the translation is to note that
the x-values in the table are 1 more than the
corresponding x-values for the parent function. The
domain is {x|x ≥ 1}, and the range is {y|y ≥ 0}.
6. SOLUTION: Make a table
x
1
1.5
2
3
y
0
1
≈ 0.7
≈ 1.4
Plot the points and draw a smooth curve. SOLUTION: Make a table.
x
2
2.5
3
4
y
0
2
≈ 1.4
≈ 2.8
Plot the points and draw a smooth curve. 4
≈ 1.7
6
4
The value 1 is being subtracted from the square root
of the parent function
, so the graph is
This graph is the result of a vertical stretch of the
translated 1 unit right from the parent graph
.
Another way to identify the translation is to note that
the x-values in the table are 1 more than the
corresponding x-values for the parent function. The
domain is {x|x ≥ 1}, and the range is {y|y ≥ 0}.
graph of y =
followed by a translation 2 units
right. The domain is {x|x ≥ 2}, and the range is {y|y
≥ 0}.
7. GEOMETRY The length of the side of a square is
given by the function
, where A is the area
of the square. What is the length of the side of a
square that has an area of 121 square inches?
A 121 inches
B 44 inches
C 11 inches
D 10 inches
6. SOLUTION: Make a table.
x
2
2.5
3
4
y
0
2
≈ 1.4
≈ 2.8
Plot the points and draw a smooth curve. 6
4
SOLUTION: The side length of the square is 11 inches. So C is
the correct choice.
Simplify each expression.
8. SOLUTION: This graph is the result of a vertical stretch of the
graph of y =
followed by a translation 2 units
right. The domain is {x|x ≥ 2}, and the range is {y|y
≥ 0}.
eSolutions Manual - Powered by Cognero
7. GEOMETRY The length of the side of a square is
given by the function
, where A is the area
9. Page 3
SOLUTION: Mid-Chapter Quiz
9. SOLUTION: 13. SATELLITES A satellite is launched into orbit 200
kilometers above Earth. The orbital velocity of a
satellite is given by the formula
. v is
velocity in meters per second, G is a given constant,
mE is the mass of Earth, and r is the radius of the
satellite’s orbit in meters.
a. The radius of Earth is 6,380,000 meters. What is
the radius of the satellite’s orbit in meters?
b. The mass of Earth is 5.97 × 1024 kilograms, and
10. SOLUTION: where N is in the constant G is
Newtons. Use the formula to find the orbital velocity
of the satellite in meters per second.
SOLUTION: a. Convert 200 kilometers to meters.
11. SOLUTION: The satellite is orbiting 200,000 meters above Earth.
The satellite’s orbit is equal to the sum of the radius
of Earth and the distance that the satellite is from
Earth.
6,380,000 m + 200,000 m = 6,580,000 m
The radius of the satellite’s orbit is 6,580,000 meters.
b. Substitute 6.77 × 10
–11
24
for G, 5.97 × 10 for mg
and 6,580,000 for r. 12. SOLUTION: The velocity of the satellite is approximately 7779
meters per second.
14. Which expression is equivalent to
eSolutions Manual - Powered by Cognero
13. SATELLITES A satellite is launched into orbit 200
kilometers above Earth. The orbital velocity of a
?
Page 4
F 16. SOLUTION: The velocity
of the satellite is approximately 7779
Mid-Chapter
Quiz
meters per second.
14. Which expression is equivalent to
?
17. SOLUTION: F G H 2
J 4
18. SOLUTION: SOLUTION: 19. SOLUTION: So, the correct choice is G.
Simplify each expression.
20. 15. SOLUTION: SOLUTION: 16. SOLUTION: 21. SOLUTION: 17. SOLUTION: eSolutions Manual - Powered by Cognero
18. 22. GEOMETRY Find the area of the rectangle.
Page 5
Mid-Chapter Quiz
22. GEOMETRY Find the area of the rectangle.
24. SOLUTION: SOLUTION: Substitute
for , and
for w. Check:
25. SOLUTION: The area of the rectangle is
square units.
Solve each equation. Check your solution.
23. SOLUTION: Check:
Check:
26. SOLUTION: 24. SOLUTION: eSolutions Manual - Powered by Cognero
Check:
Page 6
Because –4 does not satisfy the original equation, 4 is
the only solution.
Mid-Chapter Quiz
27. 26. SOLUTION: SOLUTION: Check:
Check:
Because
Because –4 does not satisfy the original equation, 4 is
the only solution.
does not satisfy the original equation, 5 is
the only solution.
28. 27. SOLUTION: SOLUTION: Check:
Check:
29. GEOMETRY The lateral surface area S of a cone
eSolutions Manual - Powered by Cognero
Because
does not satisfy the original equation, 5 is
the only solution.
can be found by using the formula
,
Page 7
where r is the radius of the base and h is the height
of the cone. Find the height of the cone.
Mid-Chapter Quiz
29. GEOMETRY The lateral surface area S of a cone
can be found by using the formula
,
where r is the radius of the base and h is the height
of the cone. Find the height of the cone.
SOLUTION: Substitute 121 for s and 3 for r. Then solve for h. The height of the cone is about 12.5 inches.
eSolutions Manual - Powered by Cognero
Page 8
across the x-axis. Another way to identify the graph i
to notice that the y-values in the table are –1 times th
corresponding y-values for the parent function. The
domain is {x|x ≥ 0} and the range is {y|y ≤ 0}.
Practice Test - Chapter 10
Graph each function, and compare to the parent
graph. State the domain and range.
2. 1. SOLUTION: Make a table.
x
0
0.5
1
2
3
y
0
≈ –
–1
≈ –
≈ –
0.7
1.4
1.7
Plot the points and draw a smooth curve.
SOLUTION: Make a table.
x
0
0.5
1
2
3
y
0
≈ 0.18 0.25 ≈ 0.35 ≈ 0.43
Plot the points and draw a smooth curve.
4
–2
The parent function
is multiplied by a value l
than 1 and greater than 0, so the the graph is a vertic
compression of
. Another way to identify the
The parent function
is multiplied by the
negative of 1, so the graph is a reflection of across the x-axis. Another way to identify the graph i
to notice that the y-values in the table are –1 times th
corresponding y-values for the parent function. The
domain is {x|x ≥ 0} and the range is {y|y ≤ 0}.
2. SOLUTION: Make a table.
x
0
0.5
1
2
3
y
0
≈ 0.18 0.25 ≈ 0.35 ≈ 0.43
Plot the points and draw a smooth curve.
compression is to notice that the y-values in the table
times the corresponding y-values for the parent functi
The domain is {x|x ≥ 0} and the range is {y|y ≥ 0}.
3. SOLUTION: Make a table.
x
0
0.5
1
2
3
y
5
6
≈ 5.7
≈ 6.4
≈ 6.7
Plot the points and draw a smooth curve.
4
7
The value 5 is being added to the parent function
, so the graph is translated up 5 units from th
The parent function
is multiplied by a value l
than 1 and greater than 0, so the the graph is a vertic
eSolutions Manual - Powered by Cognero
compression of
. Another way to identify the
compression is to notice that the y-values in the table
parent graph
. Another way to identify the
translation is to note that the y-values in the table are
greater than the corresponding y-values for the paren
function. The domain is {x|x ≥ 0} and the range is {y|
Page 1
≥ 5}.
compression of
parent graph
. Another way to identify the
translation is to note that the y-values in the table are
greater than the corresponding y-values for the paren
function. The domain is {x|x ≥ 0} and the range is {y|
≥ 5}.
. Another way to identify the
compression is to notice that the y-values in the table
timesTest
the corresponding
Practice
- Chapter 10y-values for the parent functi
The domain is {x|x ≥ 0} and the range is {y|y ≥ 0}.
3. 4. SOLUTION: Make a table.
x
0
0.5
1
2
3
y
5
6
≈ 5.7
≈ 6.4
≈ 6.7
Plot the points and draw a smooth curve.
SOLUTION: Make a table.
x
0
2
4
–4
–2
y
0
2
≈ 1.4
≈ 2.4
≈ 2.8
Plot the points and draw a smooth curve.
4
7
The value 5 is being added to the parent function
, so the graph is translated up 5 units from th
The value 4 is being added to the square root of the
parent function
, so the graph is translated 4
parent graph
. Another way to identify the
translation is to note that the y-values in the table are
greater than the corresponding y-values for the paren
function. The domain is {x|x ≥ 0} and the range is {y|
≥ 5}.
units left from the parent graph
. Another
way to identify the translation is to note that the xvalues in the table are 4 less than the corresponding
x-values for the parent function. The domain is {x|x
≥ –4}, and the range is {y|y ≥ 0}.
4. 5. GEOMETRY The length of the side of a square is
SOLUTION: Make a table.
x
0
2
4
–4
–2
y
0
2
≈ 1.4
≈ 2.4
≈ 2.8
Plot the points and draw a smooth curve.
given by the function s =
, where A is the area
of the square. What is the perimeter of a square that
has an area of 64 square inches?
A 64 inches
B 8 inches
C 32 inches
D 16 inches
SOLUTION: Substitute 64 for A.
The side length of the square is 8 inches.
The value 4 is being added to the square root of the
parent function
, so the graph is translated 4
units left from the parent graph
. Another
way to identify the translation is to note that the xvalues in the table are 4 less than the corresponding
eSolutions
Manual
by Cognero
x-values
for- Powered
the parent
function. The domain is {x|x
≥ –4}, and the range is {y|y ≥ 0}.
The perimeter of the square is 32 inches. The correct
choice is C.
Page 2
Simplify each expression.
6. units left from the parent graph
. Another
way to identify the translation is to note that the xvalues in the table are 4 less than the corresponding
x-values
the parent10
function. The domain is {x|x
Practice
Testfor- Chapter
≥ –4}, and the range is {y|y ≥ 0}.
5. GEOMETRY The length of the side of a square is
given by the function s =
, where A is the area
of the square. What is the perimeter of a square that
has an area of 64 square inches?
A 64 inches
B 8 inches
C 32 inches
D 16 inches
7. SOLUTION: SOLUTION: Substitute 64 for A.
The side length of the square is 8 inches.
8. SOLUTION: The perimeter of the square is 32 inches. The correct
choice is C.
Simplify each expression.
6. SOLUTION: 7. 9. SOLUTION: 10. GEOMETRY Find the area of the rectangle.
SOLUTION: F G 14
H eSolutions Manual - Powered by Cognero
J SOLUTION: Page 3
The area of the rectangle is
The correct choice is H.
Practice Test - Chapter 10
Solve each equation. Check your solution.
10. GEOMETRY Find the area of the rectangle.
square units. 11. SOLUTION: F G 14
Check:
H J SOLUTION: Substitute
for and
for w.
12. SOLUTION: Check:
The area of the rectangle is
The correct choice is H.
square units. Solve each equation. Check your solution.
11. SOLUTION: Because 13 does not satisfy the original equation, 3 is
the only solution.
13. PACKAGING A cylindrical container of chocolate
3
drink mix has a volume of about 162 in . The radius
of the container can be found by using the formula
, where r is the radius and h is the height.
Check:
eSolutions Manual - Powered by Cognero
If the height is 8.25 inches, find the radius of the
container.
SOLUTION: Substitute 8.25 for h.
Page 4
A length cannot be negative. The missing length is 10
units.
Because
not satisfy
Practice
Test13- does
Chapter
10 the original equation, 3 is
the only solution.
13. PACKAGING A cylindrical container of chocolate
3
drink mix has a volume of about 162 in . The radius
of the container can be found by using the formula
, where r is the radius and h is the height.
If the height is 8.25 inches, find the radius of the
container.
15. SOLUTION: SOLUTION: Substitute 8.25 for h.
A length cannot be negative. The missing length is
approximately 9.2 units.
The radius of the container is about 2.5 inches.
Find each missing length. If necessary, round to
the nearest tenth.
16. DELIVERY Ben and Amado are delivering a
freezer. The bank in front of the house is the same
height as the back of the truck. They set up their
ramp as shown. What is the length of the slanted part
of the ramp to the nearest foot?
14. SOLUTION: SOLUTION: Let L be the length of the slanted part of the ramp. L
can be found using the Pythagorean theorem:
A length cannot be negative. The missing length is 10
units.
17. Find the values of the three trigonometric ratios for
angle A.
15. SOLUTION: SOLUTION: eSolutions Manual - Powered by Cognero
Page 5
Keystrokes: [SIN] 9
15
The result is 36.86989764584.
So, m X is about 37°.
Practice Test - Chapter 10
17. Find the values of the three trigonometric ratios for
angle A.
19. LIGHTHOUSE How tall is the lighthouse?
SOLUTION: SOLUTION: Let h be the height of the lighthouse. L can be found
using trigonometric ratios:
18. Find
to the nearest degree.
SOLUTION: You know the measure of the side opposite the angle
and the hypotenuse. Use the sine ratio.
-1
Use a calculator and the [sin ] function to find the
measure of the angle. Keystrokes: [SIN] 9
15
The result is 36.86989764584.
So, m X is about 37°.
19. LIGHTHOUSE How tall is the lighthouse?
eSolutions Manual - Powered by Cognero
Page 6
The roots of the equation are
and . Choice A is the correct answer.
Preparing for Standardized Tests - Chapter 9
Read each problem. Identify what you need to
know. Then use the information in the problem
to solve.
2
1. Find the exact roots of the quadratic equation x + 5x
− 12 = 0.
A 2. The area of a triangle in which the length of the base
is 4 centimeters greater than twice the height is 80
square centimeters. What is the length of the base of
the triangle?
F −10
G 8
H 16
J 20
SOLUTION: You are asked to find the length of the base of a
triangle. Use the formula for finding the area of a
triangle. Let h = the height of the triangle and let 2h +
4 = the length of the base of the triangle.
B C D SOLUTION: You are given a quadratic equation and asked to find
the exact roots of the equation. Use the Quadratic
Formula to find the roots.
For this equation, a = 1, b = 5 and c = –12.
The dimensions of the triangle cannot be negative. So,
the height is 8 centimeters and the length of the base
is 2(8) + 4 or 20 centimeters. Choice J is the correct
answer.
3. Find the volume of the figure shown.
The roots of the equation are
and . Choice A is the correct answer.
A 18.5 cm3
3
2. The area of a triangle in which the length of the base
is 4 centimeters greater than twice the height is 80
square centimeters. What is the length of the base of
the triangle?
F −10
G 8
H 16
J 20
SOLUTION: You are asked to find the length of the base of a
eSolutions Manual - Powered by Cognero
triangle. Use the formula for finding the area of a
triangle. Let h = the height of the triangle and let 2h +
4 = the length of the base of the triangle.
B 91 cm
C 272 cm3
D 292.5 cm
3
SOLUTION: You are asked to find the volume of a rectangular
prism. Use the formula for the volume of a prism.
Page 1
The volume of the rectangular prism is 292.5 cubic
The dimensions of the triangle cannot be negative. So,
the height is 8 centimeters and the length of the base
is 2(8) +for
4 orStandardized
20 centimeters.
Choice
J is the 9correct
Preparing
Tests
- Chapter
answer.
3. Find the volume of the figure shown.
A 18.5 cm3
3
B 91 cm
C 272 cm3
D 292.5 cm
3
SOLUTION: You are asked to find the volume of a rectangular
prism. Use the formula for the volume of a prism.
The volume of the rectangular prism is 292.5 cubic
centimeters. Choice D is the correct answer.
4. Myron is traveling 263.5 miles at an average rate of
62 miles per hour. How long will it take Myron to
complete his trip?
F 5h 25 min
G 4h 15 min
H 5h 10 min
J 4h 25 min
SOLUTION: You are asked to find the time it will take to travel a
certain distance given the average rate. Use the
formula for distance.
It will take 4.25 hours or 4 hours and 15 minutes for
Myron to complete his trip. Choice G is the correct
answer.
eSolutions Manual - Powered by Cognero
Page 2
Standardized Test Practice - Cumulative, Chapter 1-10
1. Which equation below could match the graph shown
on the coordinate grid?
So, the graph of the equation in C will contain the
points (1, 5) and (4, 7).
The graph appears to contain the points (1, 1) and
(4, –1).
Therefore, the correct answer choice is B.
2. Simplify
.
F G A B C
D
SOLUTION: H J SOLUTION: From the graph, the y -intercept appears to be 3.
The y -intercept of the graphs for the equations given
in A and D are both 1. These two choices can be
eliminated. Check the values of the points at x = 1
and 4 for the equations given in B and C and
compare to the graph.
Therefore, the correct answer is H.
3. What is the area of the triangle below?
So, the graph of the equation in B will contain the
points (1, 1) and (4, –1).
A
B
C
D
SOLUTION: So, the graph of the equation in C will contain the
points
(1, 5)
and (4,by7).
eSolutions
Manual
- Powered
Cognero
The graph appears to contain the points (1, 1) and
Page 1
Standardized Test Practice - Cumulative, Chapter 1-10
Therefore, the correct answer is H.
3. What is the area of the triangle below?
Therefore, the correct answer is D.
4. The formula for the slant height c of a cone is
, where h is the height of the cone
and r is the radius of its base. What is the radius of
the cone below? Round to the nearest tenth.
A
B
C
D
SOLUTION: F 4.9
G 6.3
H 9.8
J 10.2
SOLUTION: Substitute h = 10 and c = 14 into the formula for the
slant height of a cone to find the radius of the cone.
Therefore, the correct answer is D.
4. The formula for the slant height c of a cone is
, where h is the height of the cone
and r is the radius of its base. What is the radius of
the cone below? Round to the nearest tenth.
The radius is about 9.8 units. Therefore, the correct
answer is H.
F 4.9
G 6.3
H 9.8
J 10.2
SOLUTION: Substitute h = 10 and c = 14 into the formula for the
slant height of a cone to find the radius of the cone.
5. Which of the following sets of measures could not be
the sides of a right triangle?
A (12, 16, 24)
B (10, 24, 26)
C (24, 45, 51)
D (18, 24, 30)
SOLUTION: A Since the measure of the longest side is 24, let c =
2
24, a = 12, and b = 16. Then determine whether c =
2
2
a +b .
eSolutions Manual - Powered by Cognero
Page 2
SOLUTION: A Since the measure of the longest side is 24, let c =
Standardized Test Practice - Cumulative, Chapter2 1-10
24, a = 12, and b = 16. Then determine whether c =
2
2
a +b .
Because 900 = 900, a triangle with side lengths 18,
24, and 30 is a right triangle.
Therefore, the correct answer is A.
6. Which of the following is an equation of the line
perpendicular to 4x – 2y = 6 and passing through (4,
–4)?
F
Because 576 ≠ 400, a triangle with side lengths 12,
16, and 24 is not a right triangle.
B Since the measure of the longest side is 26, let c =
2
26, a = 10, and b = 24. Then determine whether c =
2
2
a +b .
G
H
J
SOLUTION: First, find the slope of 4x – 2y = 6 by writing the
equation in slope-intercept form.
Because 676 = 676, a triangle with side lengths 10,
24, and 26 is a right triangle.
C Since the measure of the longest side is 51, let c =
2
51, a = 24, and b = 45. Then determine whether c =
2
2
a +b .
So, the slope is 2, and a line perpendicular to y = 2x –
3 will have a slope of
.
Next, find the line passing through (–4, 4).
Because 2601 = 2601, a triangle with side lengths 24,
45, and 51 is a right triangle.
D Since the measure of the longest side is 30, let c =
2
30, a = 18, and b = 24. Then determine whether c =
2
2
a +b .
Because 900 = 900, a triangle with side lengths 18,
24, and 30 is a right triangle.
Therefore, the correct answer is A.
eSolutions Manual - Powered by Cognero
6. Which of the following is an equation of the line
perpendicular to 4x – 2y = 6 and passing through (4,
Therefore, the correct answer is J.
7. The scale on a map shows that 1.5 centimeters is
equivalent to 40 miles. If the distance on the map
between two cities is 8 centimeters, about how many
miles apart are the cities?
A 178 miles
B 213 miles
C 224 miles
D 275 miles
SOLUTION: Use ratios to find the distance.
Page 3
SOLUTION: Standardized
Test Practice - Cumulative, Chapter 1-10
Therefore, the correct answer is J.
7. The scale on a map shows that 1.5 centimeters is
equivalent to 40 miles. If the distance on the map
between two cities is 8 centimeters, about how many
miles apart are the cities?
A 178 miles
B 213 miles
C 224 miles
D 275 miles
SOLUTION: Use ratios to find the distance.
10. GRIDDED RESPONSE In football, a field goal is
worth 3 points, and the extra point after a touchdown
is worth 1 point. During the 2006 season, John Kasay
of the Carolina Panthers scored a total of 100 points
for his team by making a total of 52 field goals and
extra points. How many field goals did he make?
SOLUTION: Setup and solve a system in equations where x
represents the number of field goals and y represents
the number of extra points.
First, solve for x.
The cities are about 213 miles apart. Therefore, the
correct answer is B.
8. GRIDDED RESPONSE How many times does the
2
graph of y = x - 4x + 10 cross the x-axis?
SOLUTION: 2
Since the equation y = x – 4x + 10 is a quadratic, it
will cross the x-axis 0, 1, or 2 times. Graph the equation.
Therefore, the number of field goals is 24.
11. Shannon bought a satellite radio and a subscription to
satellite radio. What is the total cost for his first year
of service?
SOLUTION: Let C represent the total cost for the first year of
service.
[–10, 10] scl: 1 by [–10, 10] scl: 1
2
The graph of y = x – 4x + 10 does not cross the xaxis. Therefore, the answer is 0.
9. Factor
completely.
SOLUTION: Therefore, the total cost is $183.87.
12. GRIDDED RESPONSE The distance required for
a car to stop is directly proportional to the square of
its velocity. If a car can stop in 242 meters at 22
kilometers per hour, how many meters are needed to
stop at 30 kilometers per hour?
SOLUTION: Substitute 242 for d and 22 for v. 10. GRIDDED RESPONSE In football, a field goal is
worth 3 points, and the extra point after a touchdown
eSolutions
Manual
- Powered
by Cognero
is worth
1 point.
During
the 2006 season, John Kasay
of the Carolina Panthers scored a total of 100 points
for his team by making a total of 52 field goals and
Page 4
Standardized
Test Practice - Cumulative, Chapter 1-10
Therefore, the total cost is $183.87.
12. GRIDDED RESPONSE The distance required for
a car to stop is directly proportional to the square of
its velocity. If a car can stop in 242 meters at 22
kilometers per hour, how many meters are needed to
stop at 30 kilometers per hour?
SOLUTION: Substitute 242 for d and 22 for v. 15. GRIDDED RESPONSE For the first home
basketball game, 652 tickets were sold for a total
revenue of $5216. If each ticket costs the same, how
much is the cost per ticket? State your answer in
dollars.
SOLUTION: Let x be the cost of a ticket.
Thus, each ticket costs $8.00.
A car will need 450 meters to stop at 30 kilometers
per hour.
13. The highest point in Kentucky is at an elevation of
4145 feet above sea level. The lowest point in the
state is at an elevation of 257 feet above sea level.
Write an inequality for this situation.
16. Karen is making a map of her hometown using a
coordinate grid. The scale of her map is 1 unit = 2.5
miles.
SOLUTION: 257 ≤ x ≤ 4,145
14. Simplify the expression below. Show your work.
SOLUTION: a. Use the Pythagorean Theorem to find the actual
distance between Karen’s school and the park.
Round to the nearest tenth of a mile
if necessary. b. Suppose Karen’s house is located at (0.5, 0.5).
What is farthest from her house, the zoo, the park,
the school, or the mall?
15. GRIDDED RESPONSE For the first home
basketball game, 652 tickets were sold for a total
revenue of $5216. If each ticket costs the same, how
much is the cost per ticket? State your answer in
dollars.
SOLUTION: Let x be the cost of a ticket.
eSolutions Manual - Powered by Cognero
SOLUTION: a. The coordinates of the school are (4, 5), and the
coordinates of the park are (5, –5). Use the
Pythagorean theorem to find the distance between
the school and the park.
Page 5
SOLUTION: a. The coordinates of the school are (4, 5), and the
coordinates of the park are (5, –5). Use the
Pythagorean
theorem
to find
the distanceChapter
between 1-10
Standardized
Test
Practice
- Cumulative,
the school and the park.
The school is located at (4, 5), so the distance
between Karen's house and the school is:
So, the distance is 10.04 units. Since 1 unit = 2.5
miles, the distance in miles is 10.04 · 2.5 or about
25.1 miles.
The park is 7.11 units away from Karen's house, the
school is 5.7, and the zoo is 4.75. The park is farthest
from Karen's house, which will be true even if the
scale is changed. b. Let (x 1, y 1) = (0.5, 0.5). The zoo is located at (–
4, 2), so the distance between Karen's house and
the zoo is:
The park is located at (5, –5), so the distance
between Karen's house and the park is:
The school is located at (4, 5), so the distance
between Karen's house and the school is:
eSolutions Manual - Powered by Cognero
Page 6
SOLUTION: In a right triangle, the side opposite the right angle is
the hypotenuse. This side is always the longest. So,
the statement is false. The longest side of a right
triangle is the hypotenuse.
Study Guide and Review - Chapter 10
State whether each sentence is true or false . If
false , replace the underlined word, phrase,
expression, or number to make a true
sentence.
1. A triangle with sides having measures of 3, 4, and 6
is a right triangle.
SOLUTION: 2
If a, b, and c are the sides of a right triangle, then c
2
2
2
= a + b , where c is the greatest number. Since 6
2
2
2
2
2
= 36 and 3 + 4 = 25, 6 ≠ 3 + 4 . Thus, a triangle
with sides having measures of 3, 4, and 6 is not a
2
2
right triangle. So, the statement is false. Since 3 + 4
2
= 25 and 5 = 25, a triangle with sides having
measures of 3, 4, and 5 would be a right triangle.
2. The expressions
and
are equivalent.
6. The cosine of an angle is found by dividing the
measure of the side opposite to the angle by the
hypotenuse.
SOLUTION: The sine of an angle is found by dividing the measure
of the side opposite of the angle by the hypotenuse.
So, the statement is false. The cosine of an angle is
found by dividing the measure of the side adjacent
to the angle by the hypotenuse.
7. The domain of the function
is .
SOLUTION: The domain of the function
the statement is false.
is {x|x ≥ 0}. So, 8. After the first step in solving
= x + 5, you
2
would have 2x + 4 = x + 10x + 25.
SOLUTION: SOLUTION: false;
To solve
= x + 5, you first need to square
each side of the equation. After the first step in
3. The expressions 2 +
and are conjugates.
SOLUTION: Binomials of the form
and are called conjugates. So, the binomials and are conjugates. The statement is
true.
4. In the expression −5
solving
= x + 5, you would have 2x + 4 = x
+ 10x + 25. The statement is true.
2
9. The converse of the Pythagorean Theorem is true.
SOLUTION: 2
2
2
If c ≠ a + b , then the triangle is not a right
triangle. So, the converse of the Pythagorean
Theorem is true. The statement is true.
, the radicand is 2.
SOLUTION: The expression under the radical sign is called the
radicand. In the expression
, the radicand is 2.
The statement is true.
5. The shortest side of a right triangle is the hypotenuse.
SOLUTION: In a right triangle, the side opposite the right angle is
the hypotenuse. This side is always the longest. So,
the statement is false. The longest side of a right
triangle is the hypotenuse.
6. The cosine of an angle is found by dividing the
measure of the side opposite to the angle by the
hypotenuse.
SOLUTION: The Manual
sine of- an
anglebyisCognero
found by dividing the measure
eSolutions
Powered
of the side opposite of the angle by the hypotenuse.
So, the statement is false. The cosine of an angle is
found by dividing the measure of the side adjacent
10. The range of the function
is .
SOLUTION: is {y|y ≥ 0}. So, The range of the function
the statement is false.
Graph each function. Compare to the parent
graph. State the domain and range.
11. y =
– 3
SOLUTION: Make a table.
x
y
0
3
0.5
≈ –2.3
1
–2
2
≈ –
1.6
3
≈ –1.3
Page 1
Plot the points on a coordinate systems and draw a
curve that connects them.
10. The range of the function
is .
SOLUTION: The
rangeand
of the
function
Study
Guide
Review
- Chapter is {y|y ≥ 0}. So, 10
the statement is false.
Graph each function. Compare to the parent
graph. State the domain and range.
11. y =
Make a table.
0
3
12. y =
+ 2
SOLUTION: Make a table.
x
0
0.5
y
2
≈ 2.7
– 3
SOLUTION: x
y
parent graph
. Another way to identify the
translation is to note that the y-values in the table are
than the corresponding y-values for the parent functi
domain is {x|x ≥ 0} and the range is {y|y ≥ –3}.
0.5
≈ –2.3
1
–2
2
≈ –
1.6
3
≈ –1.3
1
3
2
≈ 3.4
3
≈ 3.7
4
4
Plot the points on a coordinate system and draw a
smooth curve that connects then.
Plot the points on a coordinate systems and draw a
curve that connects them.
The value 2 is being added to the parent function
, so the graph is translated up 2 units from
The value 3 is being subtracted from the parent funct
, so the graph is translated down 3 units from
parent graph
. Another way to identify the
translation is to note that the y-values in the table are
than the corresponding y-values for the parent functi
domain is {x|x ≥ 0} and the range is {y|y ≥ –3}.
12. y =
13. y = −5
SOLUTION: x
0
y
0
+ 2
SOLUTION: Make a table.
x
0
0.5
y
2
≈ 2.7
the parent graph
. Another way to identify
the translation is to note that the y-values in the table
are 2 greater than the corresponding y-values for the
parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ 2}.
1
3
2
≈ 3.4
3
≈ 3.7
0.5
≈ –3.5
1
–5
2
≈ –7.1
4
4
Plot the points on a coordinate system and draw a
smooth curve that connects then.
The parent function
is multiplied by a value
less than 1, so the graph is a vertical stretch of
and a reflection across the x-axis. Another
way to identify the stretch is to notice that the yvalues in the table are –5 times the corresponding yvalues for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≤ 0}.
eSolutions Manual - Powered by Cognero
Page 2
14. y =
The value 2 is being added to the parent function
− 6
SOLUTION: the parent graph
. Another way to identify
the translation is to note that the y-values in the table
are 2 greater than the corresponding y-values for the
Study
Guide
and Review
- Chapter
10
parent
function.
The domain
is {x|x ≥ 0} and the range is {y|y ≥ 2}.
13. y = −5
15. y =
SOLUTION: x
0
y
0
0.5
≈ –3.5
1
–5
2
≈ –7.1
The parent function
is multiplied by a value
less than 1, so the graph is a vertical stretch of
and a reflection across the x-axis. Another
way to identify the stretch is to notice that the yvalues in the table are –5 times the corresponding yvalues for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≤ 0}.
14. y =
units from the parent graph
. Another way to
identify the translation is to note that the y-values in
the table are 6 less than the corresponding y-values
for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ –6}.
− 6
SOLUTION: x
0
0.5
y
–6
≈ –
5.3
1
–5
2
≈ –
4.6
3
≈ –
4.3
4
–4
units from the parent graph
. Another way to
identify the translation is to note that the y-values in
the table are 6 less than the corresponding y-values
for the parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ –6}.
15. y =
2
1
eSolutions Manual - Powered by Cognero
3
≈ 1.4
4
≈ 1.7
2
1
3
≈ 1.4
4
≈ 1.7
The value 1 is being subtracted from the square root
of the parent function
, so the graph is
translated 1 unit right from the parent graph
.
Another way to identify the translation is to note that
the x-values in the table are 1 more than the
corresponding x-values for the parent function. The
domain is {x|x ≥ 1}, and the range is {y|y ≥ 0}.
16. y =
The value 6 is being subtracted from the parent
function
, so the graph is translated down 6
SOLUTION: x
1
1.5
y
0
≈ 0.7
SOLUTION: x
1
1.5
y
0
≈ 0.7
+ 5
SOLUTION: x
0
0.5
y
5
≈ 5.7
1
6
2
≈ 6.4
3
≈ 6.7
4
7
The value 5 is being added to the parent function
, so the graph is translated up 5 units from
the parent graph
. Another way to identify
the translation is to note that the y-values in the table
are 5 greater than the corresponding y-values for the
parent function. The domain is {x|x ≥ 0} and the range is {y|y ≥ 5}.
17. GEOMETRY The function s =
can be used to
find the length of a side of a square given its area.
Use this function to determine the length of a side of
a square with an area of 90 square inches. Round to
the nearest tenth if necessary.
SOLUTION: Page 3
the parent graph
. Another way to identify
the translation is to note that the y-values in the table
are 5 greater than the corresponding y-values for the
Study
Guide
and Review
- Chapter
10
parent
function.
The domain
is {x|x ≥ 0} and the range is {y|y ≥ 5}.
17. GEOMETRY The function s =
can be used to
find the length of a side of a square given its area.
Use this function to determine the length of a side of
a square with an area of 90 square inches. Round to
the nearest tenth if necessary.
22. SOLUTION: SOLUTION: 23. SOLUTION: The side length of the square is about 9.5 inches.
Simplify.
18. SOLUTION: 24. SOLUTION: 19. SOLUTION: 20. SOLUTION: 25. SOLUTION: 21. SOLUTION: 22. eSolutions Manual - Powered by Cognero
SOLUTION: Page 4
Study Guide and Review - Chapter 10
27. 25. SOLUTION: SOLUTION: 28. WEATHER To estimate how long a thunderstorm
will last, use
, where t is the time in hours
and d is the diameter of the storm in miles. A storm
is 10 miles in diameter. How long will it last?
SOLUTION: Substitution 10 for d. 26. SOLUTION: The storm will last about 2.15 hours. To convert 2.15
hours to hours and minutes, multiply the number of
minutes in an hour by the decimal part. Because 60 • 0.15 = 9, 2.15 hours is equal to 2 hours and 9
minutes.
Simplify each expression.
29. SOLUTION: 27. SOLUTION: 30. SOLUTION: eSolutions Manual - Powered by Cognero
Page 5
SOLUTION: Study Guide and Review - Chapter 10
36. MOTION The velocity of a dropped object when it
30. hits the ground can be found using
, where
v is the velocity in feet per second, g is the
acceleration due to gravity, and d is the distance in
feet the object drops. Find the speed of a penny
when it hits the ground, after being dropped from 984
feet. Use 32 feet per second squared for g.
SOLUTION: SOLUTION: Substitute 32 for g and 984 for d. 31. SOLUTION: 32. SOLUTION: The speed of the penny when it hits the ground is
about 250.95 feet per second.
Solve each equation. Check your solution.
33. 37. SOLUTION: SOLUTION: 34. SOLUTION: Because the square root of a number cannot be
negative, there is no solution.
38. SOLUTION: 35. SOLUTION: 36. MOTION The velocity of a dropped object when it
hits the ground can be found using
, where
v is the velocity in feet per second, g is the
acceleration due to gravity, and d is the distance in
feet the object drops. Find the speed of a penny
whenManual
it hits- Powered
the ground,
after being dropped from 984
eSolutions
by Cognero
feet. Use 32 feet per second squared for g.
SOLUTION: Check:
Page 6
Because the square root of a number cannot be
Study
Guide there
and Review
- Chapter 10
negative,
is no solution.
38. 40. SOLUTION: SOLUTION: Check:
Check:
41. SOLUTION: 39. SOLUTION: Check:
Check:
Because 5 does not satisfy the original equation, 12 is
the only solution.
40. SOLUTION: 42. SOLUTION: eSolutions Manual - Powered by Cognero
Check:
Page 7
Check:
Because
doesReview
not satisfy
the original
Study
Guide5and
- Chapter
10 equation, 12 is
the only solution.
The skydiver will fall 1600 feet before opening the
parachute.
Determine whether each set of measures can
be lengths of the sides of a right triangle.
44. 6, 8, 10
42. SOLUTION: SOLUTION: Since the measure of the longest side is 10, let c =
2
10, a = 6, and b = 8. Then determine whether c =
2
2
a +b .
Check:
2
2
2
Yes, because c = a + b , a triangle with side
lengths 6, 8, and 10 is a right triangle.
45. 3, 4, 5
Because 10 does not satisfy the original equation, 5 is
the only solution.
43. FREE FALL Assuming no air resistance, the time t
in seconds that it takes an object to fall h feet can be
determined by
SOLUTION: Since the measure of the longest side is 5, let c = 5, a
2
2
= 3, and b = 4. Then determine whether c = a +
2
b .
. If a skydiver jumps from an
airplane and free falls for 10 seconds before opening
the parachute, how many feet does she fall?
SOLUTION: Substitute 10 for t. 2
2
2
Yes, because c = a + b , a triangle with side
lengths 3, 4, and 5 is a right triangle.
46. 12, 16, 21
SOLUTION: Since the measure of the longest side is 21, let c =
2
21, a = 12, and b = 16. Then determine whether c =
2
2
a +b .
2
The skydiver will fall 1600 feet before opening the
parachute.
Determine whether each set of measures can
right triangle.
44. 6, 8, 10
eSolutions
Manual - Powered
by Cognero
be lengths
of the sides
of a
SOLUTION: 2
2
No, because c ≠ a + b , a triangle with side lengths
12, 16, and 21 is not a right triangle.
47. 10, 12, 15
SOLUTION: Since the measure of the longest side is 15, let c Page
= 8
2
15, a = 10, and b = 12. Then determine whether c =
2
2
a +b .
2
2
2
No,
because
, a triangle
Study
Guide
andc Review
10with side lengths
≠ a + -bChapter
12, 16, and 21 is not a right triangle.
47. 10, 12, 15
2
a +b .
2
2
2
No, because c ≠ a + b , a triangle with side lengths
10, 12, and 15 is not a right triangle.
48. 2, 3, 4
2
SOLUTION: Since the measure of the longest side is 13, let c =
2
13, a = 5, and b = 12. Then determine whether c =
2
2
a +b .
2
2
2
Yes, because c = a + b , a triangle with side
lengths 5, 12, and 13 is a right triangle.
51. 15, 19, 23
SOLUTION: Since the measure of the longest side is 4, let c = 4, a
2
2
= 2, and b = 3. Then determine whether c = a +
2
b .
2
2
2
No, because c ≠ a + b , a triangle with side lengths
2, 3, and 4 is not a right triangle.
49. 7, 24, 25
SOLUTION: Since the measure of the longest side is 25, let c =
2
25, a = 7, and b = 24. Then determine whether c =
2
2
50. 5, 12, 13
SOLUTION: Since the measure of the longest side is 15, let c =
2
15, a = 10, and b = 12. Then determine whether c =
2
2
Yes, because c = a + b , a triangle with side
lengths 7, 24, and 25 is a right triangle.
2
a +b .
2
2
SOLUTION: Since the measure of the longest side is 23, let c =
2
23, a = 15, and b = 19. Then determine whether c =
2
2
a +b .
2
2
2
No, because c ≠ a + b , a triangle with side lengths
15, 19, and 23 is not a right triangle.
52. LADDER A ladder is leaning on a building. The
base of the ladder is 10 feet from the building, and
the ladder reaches up 15 feet on the building. How
long is the ladder?
SOLUTION: Use the Pythagorean Theorem, substituting 10 for a
and 15 for b.
2
Yes, because c = a + b , a triangle with side
lengths 7, 24, and 25 is a right triangle.
50. 5, 12, 13
The ladder is approximately 18.0 feet long.
SOLUTION: Since the measure of the longest side is 13, let c =
2
13, a = 5, and b = 12. Then determine whether c =
2
2
eSolutions
a +Manual
b . - Powered by Cognero
Find the values of the three trigonometric
ratios for angle A .
Page 9
Study Guide and Review - Chapter 10
The ladder is approximately 18.0 feet long.
Find the values of the three trigonometric
ratios for angle A .
55. RAMPS How long is the ramp?
53. SOLUTION: SOLUTION: You know the measure of the side opposite the angle
and the measure of the angle. Use the sine ratio.
The ramp is 6 feet long.
54. SOLUTION: 55. RAMPS How long is the ramp?
eSolutions Manual - Powered by Cognero
SOLUTION: You know the measure of the side opposite the angle
Page 10
