Download Chapter 8 Study Guide - Problems involving Equations and

Chapter 8 Study Guide - Problems involving Equations and Inequalities
1. Carolee makes quilts to sell at an arts and crafts fair. She charges $75 per quilt.
a.
Name the quantity that is constant.
b.
Which quantity depends on the other?
c.
Write an algebraic equation to represent this situation. Define your variables. Use your equation to
determine how much Carolee will earn if she sells 15 quilts.
.
2. Meredith is having a deck built onto her house. The table shows the possible dimensions of her deck.
Width (in feet)
10
11
12
13
14
15
16
Length (in feet)
16.5
18.5
20.5
22.5
24.5
26.5
28.5
a.
Let l represent the length of the deck. Write an equation that represents the length of the deck in terms
of the width, w.
b.
.
Calculate the length of the deck if the width is 20 feet.
c.
.
If the length of a deck is 25 feet long, what is the width?
d.
.
Calculate the area of a deck that measures 15.5 feet wide.
3. One event in the winter Olympics is the 30-kilometer Nordic skiing event. In 2006, Katerina Neumannova of
the Czech Republic won the event in 1 hour, 22 minutes, and 25.4 seconds.
a.
.
How fast in meters per minute did Neumannova average? Explain.
b.
Imagine you are watching the 2006 race and Neumannova is at the halfway mark. How many meters
has she skied?
.
c.
If Neumannova is skiing at the average speed you determined in part (a), how many meters will she
travel in a quarter of an hour?
4. In a freefall skydive, a skydiver begins at an altitude of 13,000 feet. During the one-minute freefall, the
skydiver drops towards Earth at a rate of 176 feet per second.
a.
Identity the two quantities that are changing, identify the independent and dependent quantities,
define variables for those quantities, and write an equation to represent a skydiver’s falling distance.
.
b.
In this problem, what is the unit rate of change?
c.
Draw the graph to represent the problem situation.
5. To earn some extra money, Ben mows lawns for people in his neighborhood. He earns $18 per lawn.
a.
Name the quantity that is constant.
b.
Which quantity depends on the other?
c.
Write an algebraic equation to represent this situation. Define your variables. Use your equation to
determine how much Ben will earn for 21 lawns.
.
6. Solve
.
7. The winner of the 2008 Iditarod Trail Sled Dog Race was Lance Mackey. The course is 1112 miles long. His
time was 9 days, 11 hours, 46 minutes, and 48 seconds.
a.
What was Mackey’s average speed in miles per hour? Explain.
b.
Imagine you are watching the 2008 race and Mackey is at the one-quarter mark. How many miles has
he traveled?
c.
If Mackey is traveling at the average speed you determined in part (a), how many miles will he travel
in half a day?
8. Jerome is a photographer. He earns $125 per hour.
a.
Name the quantity that is constant.
b.
Which quantity depends on the other?
c.
Complete the table that represents various numbers of hours worked.
Number of Hours
0
1
5
7
250
Earnings (in dollars)
10
1125
d.
Create a graph of the data from the table.
e.
Write an algebraic equation to represent this situation. Define your variables.
.
f.
Use your equation to determine how much Jerome will earn if he works 12 hours.
9. Sadiq makes tables with a mosaic top. He earns $125 per table sold.
a.
Name the quantity that is constant.
b.
Which quantity depends on the other?
c.
Complete the table that represents various numbers of tables that Sadiq has sold.
Number of Tables
0
1
Earnings (in dollars)
d.
Create a graph of the data from the table.
5
250
13
875
1250
e.
Write an algebraic equation to represent this situation. Define your variables. Use your equation to
determine how much Sadiq will earn if he sells 22 tables.
10. The 2008 winner of the Tour de France was Carlos Sastre. He biked the 2212-mile race in 87 hours, 52
minutes, and 52 seconds.
a.
.
What was Sastre’s average speed in miles per hour? Explain.
b.
Sara traveled to Europe to watch the Tour de France in 2008. Her plane was delayed, so she arrived
late to the race. Carlos Sastre was at the one-quarter mark when Sara arrived. How many miles had he
biked?
.
c.
If Sastre is biking at the average speed you determined in part (a), how many miles will he travel in
half an hour?
.
d.
Consider the situation after Sara arrived and the distance that Sastre traveled. What are the two
quantities that are changing? Which quantity depends on the other? Define and identify the
independent and dependent variables for these quantities with their units of measure.
.
e.
Complete the table for the given values.
Quantity
Unit of Measure
Variable
Time After Sara Arrives
Distance
Hours
Miles
t
d
0
2
10
50
55
60
65
66
f.
Write an equation or rule for calculating the value of the dependent variable when the value of the
independent variable is given.
11. Gene is landing his glider. He is at 10,450 feet and beginning his descent. He descends at a rate of 950 feet
per minute.
a.
Identify the two quantities that are changing, identify the independent and dependent quantities,
define variables for those quantities, and write an equation to represent the glider’s altitude.
.
b.
In this problem, what is the unit rate of change?
.
c.
What would the altitude of the glider be after
.
d.
At what time would the glider be at an altitude of 5000 feet?
.
Solve each equation and verify your solution.
12.
.
13.
.
minutes? 45 seconds?
14.
.
15.
16. The graph shows the relationship between two quantities.
a.
Complete the table using the points shown in the graph.
x
b.
y
What is the unit rate of change? Explain.
.
c.
.
Write an equation for this relationship.
d.
Write a problem situation that fits the equation.
Complete each table and then create a graph from the table of values.
17. A children’s museum charges $10 admission for the first visitor and $5 for each additional visitor in a group.
Number of Visitors
Cost of First Visitor
(dollars)
Total Cost
(dollars)
1
3
7
$60
$75
Write an equation that represents each problem situation. Then use the equation to answer the question.
18. The daycare center charges $120 for one week of care. Families with multiple children pay $95 for each
additional child per week. Write an equation for the total cost for one week of care in terms of the number of
children. Define your variables. How much will one week of care cost a family with 3 children?
.
19. The Dahlia plants at the garden center are 6 inches tall and grow at a rate of 2.5 inches per week. Write an
equation that represents the height of the Dahlia plants, h, in terms of the number of weeks, w.
Evaluate the two equations given in each problem situation to make a decision.
20. Manuel is planning an anniversary dinner party for his parents. He is comparing restaurants to find the most
affordable location. The total cost for dinner at Luigi’s, t, in terms of the number of guests, g, is represented
by the equation
. The total cost for dinner at Mario’s, t, in terms of the number of guests,
g, is represented by the equation
. Calculate the total cost at each restaurant if Manuel
invites 20 guests. Which restaurant is the most affordable for 20 guests?
.
Write an equation for each situation, define your variable, and solve the equation.
Then verify your solution.
21. Petit Chat cat salon charges $35 for each cat-grooming session. Pretty Kitty cat salon charges a one-time fee
of $20 and $25 for each cat-grooming session. How many grooming sessions would make their fees equal?
.
22. The Beach Shack rents boats for $60 the first hour and $30 each hour after that. The Surf Shop rents boats for
$40 an hour. At what number of hours are the rental fees for a boat from each shop equal?
Define the independent and dependent variables and write an equation to represent each problem situation.
Use the equation to complete the table.
23. A hot air balloon is rising at a rate of 10 meters per second. Write an equation that represents the height of the
balloon as it takes off from the ground.
Independent Quantity
Label
Units
0
1
2
3
Dependent Quantity
4
24. A submarine is diving at a rate of 0.25 meter per second. Write an equation that represents the depth of the
submarine as it dives below the surface.
Independent Quantity
Dependent Quantity
Label
Units
0
30
60
90
120
Use the given information to answer each question.
25. A diver is diving at a rate of 0.5 meter per second. Write an equation that represents the depth of the diver as
she dives below the surface. How deep will the diver be after 30 seconds?
26. A submarine is diving at a rate of 0.65 foot per second. Write an equation that represents the depth of the
submarine as it dives below the surface. How long will it take for the submarine to reach its goal depth of
195 feet?
.
27. A submarine is diving at a rate of 0.45 foot per second. Its goal depth is 94.5 feet. Write an inequality that
represents the time the submarine is below the surface until it reaches its goal depth. How long will the
submarine be below sea level before it reaches its goal depth of 94.5 feet?
.
28. Fernando is using a garden hose to fill his backyard pool at a rate of 10 gallons per minute. The pool already
contains 9000 gallons of water. The capacity of the pool is 12,000 gallons. Define the independent and
dependent variables and the unit rate of change. Write an equation that represents the amount of water in the
pool as it is being filled. How many gallons of water will be in the pool after 2.5 hours?
.
29. The graph shows the relationship between two quantities. Complete the table. Define the independent and
dependent variables and the unit rate of change. Write an equation that represents the data shown.
x
y
Chapter 8 Study Guide - Problems involving Equations and Inequalities
Answer Section
1. ANS:
a.
The amount Carolee charges per quilt, $75, is constant.
b.
The amount Carolee earns depends on the number of quilts she sells.
c.
, where e represents total earnings, in dollars, and q represents the number of quilts made
Carolee will earn $1125 for 15 quilts.
PTS: 1
TOP: Pre Test
2. ANS:
a.
REF: 8.1
NAT: 7.EE.3 | 7.EE.4.a
b.
A deck that is 20 feet wide will be 36.5 feet long.
c.
A deck that is 25 feet long will be 14.25 feet wide.
d.
The area of a deck that is 15.5 feet wide will be 426.25 square feet.
PTS: 1
3. ANS:
REF: 8.2
NAT: 7.EE.4.a
TOP: Pre Test
a.
Neumannova skied at an average rate of approximately 363.97 meters per minute.
b.
She has skied half way or 15,000 meters.
c.
In a quarter of an hour, she will ski 5459.6 meters.
PTS: 1
TOP: Pre Test
4. ANS:
REF: 8.5
NAT: 7.EE.3 | 7.EE.4.a | 7.EE.4.b
KEY: unit rate of change
a.
The two quantities that are changing are the time and the falling distance.
Independent: Let t  the time of the fall in seconds
Dependent: Let d  the falling distance in feet
b.
The unit rate of change is
.
c.
PTS: 1
REF: 8.6
NAT: 7.EE.3 | 7.EE.4.a
TOP: Pre Test
5. ANS:
a.
The amount Ben earns per lawn, $18, is constant.
b.
c.
The amount Ben earns depends on the number of lawns mowed.
, where t represents total earnings, in dollars, and s represents the number of lawns mowed
Ben will earn $378 for mowing 21 lawns.
PTS: 1
TOP: Post Test
6. ANS:
REF: 8.1
NAT: 7.EE.3 | 7.EE.4.a
PTS: 1
TOP: Post Test
7. ANS:
REF: 8.1
NAT: 7.EE.3 | 7.EE.4.a
a.
Mackey traveled at an average rate of about 4.88 miles per hour.
b.
He has traveled one-fourth of the way, or 278 miles.
c.
In half a day, he will travel 58.56 miles.
PTS: 1
REF: 8.5
NAT: 7.EE.3 | 7.EE.4.a | 7.EE.4.b
TOP: Post Test
KEY: unit rate of change
8. ANS:
a.
The amount Jerome earns per hour, $125, is constant.
b.
The amount Jerome earns depends on the number of hours he works.
c.
Number of Hours
0
1
2
5
7
Earnings (in dollars)
0
125
250
725
875
9
10
1125 1250
d.
e.
, where d represents total earnings, in dollars, and h represents the number of hours worked.
f.
Jerome will earn $1500 for working 12 hours.
PTS: 1
REF: 8.1
NAT: 7.EE.3 | 7.EE.4.a
TOP: Mid Ch Test
9. ANS:
a.
The amount Sadiq earns per table, $125, is constant.
b.
The amount Sadiq earns depends on the number of tables he sells.
c.
d.
Number of Tables
0
1
2
5
7
10
13
Earnings (in dollars)
0
125
250
625
875
1250
1625
e.
, where d represents total earnings, in dollars, and t represents the number of tables sold
Sadiq will earn $2750 for selling 22 tables.
PTS: 1
TOP: End Ch Test
10. ANS:
REF: 8.1
NAT: 7.EE.3 | 7.EE.4.a
a.
Sastre’s average speed was about 25.17 miles per hour.
b.
He biked one-fourth of the distance, or 553 miles.
c.
In half an hour, he will bike about 12.59 miles.
d.
The two quantities that are changing are the time after Sara has arrived and Sastre’s distance from the
starting point. The distance depends on the time. The independent variable is time, t, in hours after
Sara has arrived. The dependent variable is distance, d, in miles.
e.
Quantity
Unit of Measure
Time After Sara Arrives
Distance
Hours
Miles
t
d
0
553
2
603.34
10
804.7
50
1811.5
55
1937.35
60
2063.2
65
2189.05
Variable
66
2214.22
f.
PTS: 1
TOP: End Ch Test
11. ANS:
a.
REF: 8.5
NAT: 7.EE.3 | 7.EE.4.a | 7.EE.4.b
KEY: unit rate of change
The two quantities that are changing are the time and the altitude.
Independent: Let t  the time of the descent in minutes
Dependent: Let a  the altitude in feet
b.
The unit rate of change is
.
c.
After
minutes, the altitude would be 6175 feet.
After 45 seconds, the altitude would be 9737.5 feet.
d.
The glider would be at 5000 feet after about 5.7368 minutes.
PTS: 1
TOP: End Ch Test
12. ANS:
REF: 8.5
NAT: 7.EE.3 | 7.EE.4.a | 7.EE.4.b
KEY: unit rate of change
Verify the solution by substituting 2 for x in the equation.
PTS: 1
REF: 8.3
TOP: Skills Practice
13. ANS:
NAT: 7.EE.3 | 7.EE.4.a
KEY: multiplicative inverse | multiplying by the reciprocal
Verify the solution by substituting 12 for x in the equation.
PTS: 1
REF: 8.3
TOP: Skills Practice
14. ANS:
NAT: 7.EE.3 | 7.EE.4.a
KEY: multiplicative inverse | multiplying by the reciprocal
Verify the solution by substituting 18 for x in the equation.
PTS: 1
REF: 8.3
TOP: Skills Practice
NAT: 7.EE.3 | 7.EE.4.a
KEY: multiplicative inverse | multiplying by the reciprocal
15. ANS:
Verify the solution by substituting 4 for x in the equation.
PTS: 1
REF: 8.3
TOP: Skills Practice
16. ANS:
a.
NAT: 7.EE.3 | 7.EE.4.a
KEY: multiplicative inverse | multiplying by the reciprocal
x
y
0
18
0.5
17
1
16
1.5
15
2
14
3
12
5
8
8
2
9
b.
0
The unit rate of change is 2. For every increase in the x-value by 1, the y-value decreases by 2.
c.
d.
Answers will vary.
Sample answer: A hiker is returning to base camp on an 18-mile trail at 2 miles per hour.
PTS: 1
TOP: End Ch Test
17. ANS:
REF: 8.6
NAT: 7.EE.3 | 7.EE.4.a
Number of Visitors
Cost of First Visitor
(dollars)
Total Cost
(dollars)
1
$10
$10
3
$10
$20
7
$10
$40
11
$10
$60
14
$10
$75
PTS: 1
REF: 8.1
NAT: 7.EE.3 | 7.EE.4.a
TOP: Skills Practice
18. ANS:
Let t represent the total cost for one week of care.
Let c represent the number of children.
One week of care for a family with 3 children will cost $310.
PTS: 1
REF: 8.1
TOP: Skills Practice
19. ANS:
NAT: 7.EE.3 | 7.EE.4.a
PTS: 1
REF: 8.2
20. ANS:
Total Cost at Luigi’s
NAT: 7.EE.4.a
TOP: Skills Practice
Total Cost at Mario’s
For 20 guests, Mario’s is the most affordable.
PTS: 1
21. ANS:
REF: 8.2
NAT: 7.EE.4.a
Let x represent the number of grooming sessions.
TOP: Skills Practice
The fees would be equal at 2 grooming sessions.
Verify the solution by substituting 2 for x in the equation.
PTS: 1
REF: 8.3
TOP: Skills Practice
22. ANS:
NAT: 7.EE.3 | 7.EE.4.a
KEY: multiplicative inverse | multiplying by the reciprocal
Let h represent the number of rental hours.
The rental fees from the shops are equal at 3 hours.
Verify the solution by substituting 3 for h in the equation.
PTS: 1
REF: 8.3
TOP: Skills Practice
23. ANS:
Independent variable:
t  time in seconds
NAT: 7.EE.3 | 7.EE.4.a
KEY: multiplicative inverse | multiplying by the reciprocal
Dependent variable:
h  height in meters
Independent Quantity
Dependent Quantity
Label
Time
Height
Units
seconds
meters
0
0
PTS: 1
REF: 8.5
TOP: Skills Practice
24. ANS:
Independent variable:
t  time in seconds
1
10
2
20
3
30
4
40
NAT: 7.EE.3 | 7.EE.4.a | 7.EE.4.b
KEY: unit rate of change
Dependent variable:
d  depth in meters
Independent Quantity
Dependent Quantity
Label
Time
Depth
Units
seconds
meters
0
0
30
7.5
60
15
90
22.5
120
30
PTS: 1
REF: 8.5
TOP: Skills Practice
25. ANS:
Independent variable: t  time in seconds
Dependent variable: d  depth in meters
NAT: 7.EE.3 | 7.EE.4.a | 7.EE.4.b
KEY: unit rate of change
The diver will be 15 meters below the surface.
PTS: 1
REF: 8.5
TOP: Skills Practice
26. ANS:
Independent variable: t  time in seconds
NAT: 7.EE.3 | 7.EE.4.a | 7.EE.4.b
KEY: unit rate of change
Dependent variable: d  depth in feet
300 seconds  5 minutes
The submarine will take 5 minutes to reach its goal depth.
PTS: 1
REF: 8.5
TOP: Skills Practice
27. ANS:
Independent variable: t  time in seconds
NAT: 7.EE.3 | 7.EE.4.a | 7.EE.4.b
KEY: unit rate of change
Dependent variable: d  depth in feet
210 seconds  3.5 minutes
The submarine will be below the surface for about 3.5 minutes before reaching its goal depth.
PTS: 1
REF: 8.5
TOP: Skills Practice
28. ANS:
Independent variable: t  time in minutes
NAT: 7.EE.3 | 7.EE.4.a | 7.EE.4.b
KEY: unit rate of change
Dependent variable: w  amount of water in gallons
Unit rate of change  10 gallons per minute
There will be 10,500 gallons of water in the pool.
PTS: 1
REF: 8.6
TOP: Skills Practice
29. ANS:
x
y
0
15
1
20
2
25
3
30
4
35
NAT: 7.EE.3 | 7.EE.4.a
Independent variable x
Dependent variable y
Unit rate of change  5
PTS: 1
REF: 8.6
TOP: Skills Practice
NAT: 7.EE.3 | 7.EE.4.a