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Math 20-2
Final Review
Name:
Class:
Date of Final:
Unit 1: Quadratics
Vocabulary:
Quadratic Relation: a relation that can be written in standard form
𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐, where 𝑎 ≠ 0; for example, 𝑦 = 4𝑥² + 2𝑥 + 1.
Vertex: The point at which the quadratic function reaches its maximum or
minimum value.
Axis of symmetry: A line that seperates a 2-D figure into two identical parts.
For example, a parabola has a vertical axis of symmetry passing through its
vertex.
Maximum value: The greatest value of the dependent variable in a relation.
Zero: in a function, a value of the variable that makes the value of the
function equal to zero.
Quadratic equation: a polynomial equation of the second degree; the
standard form of the quadratic equation is 𝑎𝑥² + 𝑏𝑥 + 𝑐 = 0.
Quadratic Formula: a formula for determining the roots of a quadratic
equation in the form 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0, where a≠ 0; the quadratic formula is
written using the coefficients of the variables and the constant in the
quadratic equation that is being solved:
−𝑏 ± √𝑏 2 − 4𝑎𝑐
2𝑎
This formula is derived from 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0 by isolating x.
Inadmissible solution: a root of a quadratic equation that does not lead to a
solution that satisfies the original problem.
Math 20-2 Final Review
2
Unit 1: Quadratics
Recall:
Standard form: 𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
Factored form: 𝑓(𝑥) = 𝑎(𝑥 − 𝑟)(𝑥 − 𝑠)
Vertex form: 𝑓(𝑥) = 𝑎(𝑥 − ℎ)2 + 𝑘
Key Ideas:





The degree of all quadratic functions is 2
The standard form of a quadratic function is 𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
The graph of any quadratic function is a parabola with a single vertical
line of symmetry
A quadratic function that is written in standard form, 𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
has the following characteristics:
The highest or lowest point on the graph of the quadratic function
lies on its vertical line of symmetry
- If a is positive, the parabola opens up. If a is negative, the
parabola opens down.
- Changing the value of b changes the location of the parabola’s line
of symmetry (positive moves left, negative moves right)
- The constant term, c, is the value of the parabola’s y-intercept
When a quadratic function is written in factored form:
y = a(x – r)(x – s)
each factor can be used to determine a zero of the function by setting
each factor equal to zero and solving.
-
-
The zeros of a quadratic function correspond to the x-intercepts of
the parabola that is defined by the function
If a parabola has one or two x-intercepts, the equation of the
parabola can be written in factored form using the x-intercepts and
the coordinates of one other point on the parabola
Quadratic functions without any zeros cannot be written in factored
form
Math 20-2 Final Review
3
Unit 1: Quadratics
Practice Questions:
Multiple Choice
Identify the choice that best completes the statement or answers the
question.
y
5
____ 1. Which set of data is correct for the graph to the right?
4
3
2
A.
B.
C.
Axis of Symmetry
x = –2
x = –0.25
x = –0.5
D.
x=3
Vertex
(–0.25, –3.125)
(–0.25, 3.125)
(–0.5, 3)
(3, –0.5)
Domain
xR
xR
–2.5  x 
1.5
–3  x  2
1
Range
yR
y  3.125
y3
–5
–4
–3
–2
–1
–1
1
2
3
4
x
5
–2
–3
y5
–4
–5
____ 2. What is the correct quadratic function for the parabola to the right?
y
5
a.
b.
c.
d.
f(x)
f(x)
f(x)
f(x)
=
=
=
=
4
(x + 1)(x + 3)
(1 – x)(3 – x)
(x – 1)(x + 3)
–(x + 1)(x – 3)
3
2
1
–5
–4
–3
–2
–1
–1
1
2
–2
–3
–4
–5
____ 3. How many zeros does f(x) = (x – c)2 + d have if d > 0?
a. 1
b. 0
c. 2
d. It is impossible to determine.
____ 4. Solve 4p2 + 15p = –9 by factoring.
a. p = – , p = –3
b. p = 4, p = 3
c. p = –4, p = 3
d. p = – , p = 3
Math 20-2 Final Review
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Unit 1: Quadratics
3
4
5
x
____ 5. Solve x2 – 2x = 4 using the quadratic formula.
a.
b.
c.
d.
x
x
x
x
=
=
=
=
1+
–1 +
–1 +
1+
,x=1–
, x = –1 –
, x = –1 –
,x=1–
Written Response
6. Fill in the table for the relation y = –2x2 + 2x – 1.
Maximum or minimum
Axis of symmetry
Vertex
7. Determine the equation that defines a quadratic function with x-intercepts
located at (–9, 0) and (–2, 0) and a y-intercept of (0, 18). Provide a sketch
to support your work.
8. a) Sketch the graph of y = (x + 1)(x – 7).
b) State the maximum or minimum value of the function.
c) Express the function in standard form.
Math 20-2 Final Review
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Unit 1: Quadratics
9. Determine the quadratic function that defines the parabola in vertex form.
y
16
14
12
10
8
6
4
2
–4
–2
–2
2
4
6
8
10
12
14
16
x
–4
10. The underside of a concrete underpass forms a parabolic arch. The arch is
32 m wide at the base and 11.5 m high in the centre. What would be the
minimum headroom on a sidewalk that is built 1.5 m from the base of the
underpass?
11. A ball is thrown into the air from a bridge that is 15 m above a river. The
function that models the height, h(t), in metres, of the ball over time, t, in
seconds is
h(t) = –4.9t2 + 9t + 15
Answer each of the following questions by writing the corresponding
quadratic equation and solving the equation by graphing.
a) When is the ball 17 m above the water?
b) When does the ball hit the water?
Math 20-2 Final Review
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Unit 1: Quadratics
12. Solve 2x2 + 4x – 2.5 = 0.5x2 + 5x + 4.
13. Tori sells posters to stores. The profit function for her business is P(n) = –
0.3n2 + 4n – 5, where n is the number of posters sold per month, in
hundreds, and P(n) is the profit, in thousands of dollars.
a) How many posters must Tori sell per month to break even?
b) If Tori wants to earn a profit of $6000 (P(n)= 6), how many posters must
she sell?
14. Ty is an artist. He wants the matte around each of his square photographs to
be 6.5 cm wide. He also wants the area of the matte to be twice the area of
each photograph. What should the dimensions of each photograph be, to the
nearest tenth of a centimetre? Use a labelled diagram to solve the problem.
Math 20-2 Final Review
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Unit 1: Quadratics
15. A small movie theatre sells tickets for $15. At this price, the theatre sells
200 tickets every show. The owners know from past years that they will sell
8 more tickets per show for each price decrease of $0.50.
a) What function, E(x), can be used to model the owners’ earnings, if x
represents the price decrease in dollars?
b) What lower price would let the owners earn the same amount of money
they earn now?
c) What should the owners charge per ticket to earn the maximum amount
of money?
Math 20-2 Final Review
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Unit 1: Quadratics
Unit 2: Logic
Vocabulary:
Conjecture: A testable expression that is based on available evidence but is
not yet proved.
Inductive reasoning: Draw a general conclusion by observing patterns and
identifying properties in specific examples.
Deductive Reasoning: Drawing a specific conclusion through logical
reasoning by starting with general assumptions that are known to be valid.
Invalid Proof: A proof that contains an error in reasoning or that contains
invalid assumptions.
Circular Reasoning: An argument that is incorrect because it makes use of
the conclusion to be proved.
Practice Questions:
Multiple Choice
Identify the choice that best completes the statement or answers the
question.
____ 1. Ginerva made the following conjecture: The square of a number is always
greater than the number. Is the following equation a counterexample to this
conjecture? Explain.
52 = 25
a.
b.
c.
d.
Yes, it is a counterexample, because 25 is greater than 5.
No, it is not a counterexample, because 25 is greater than 5.
No, it is not a counterexample, because 25 is less than 5.
Yes, it is a counterexample, because 5 is less than 25.
____ 2. What type of error, if any, occurs in the following deduction?
All swimmers can swim one kilometre without stopping.
Joan is a swimmer.
Therefore, Joan can swim one kilometre without stopping.
a.
b.
c.
d.
a false assumption or generalization
an error in reasoning
an error in calculation
There is no error in the deduction.
Math 20-2 Final Review
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Unit 2: Logic
____ 3. Which number should appear in the centre of Figure 4?
a.
b.
c.
d.
Figure 1
6
120
15
240
Figure 2
Figure 3
Figure 4
Short Answer
4. Examine the following example of deductive reasoning. Why is it faulty?
Given: At 11:00 p.m. this evening, there will be a newscast on Channel 20.
There is a newscast on Channel 20 starting right now.
Deduction: It is now 11:00.
5. The square of an odd integer is subtracted from the square of an even
integer.
Develop a conjecture about whether the difference is odd or even.
Provide evidence to support your conjecture.
Math 20-2 Final Review
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Unit 2: Logic
6. Tyler made the following conjecture:A polygon with four right angles must
be a rectangle.
Matthew disagreed with Tyler’s conjecture, however, because the following
figure has four right angles, and it is not a rectangle.
How could Tyler’s conjecture be improved? Explain the changes you would
make.
7. What type of error occurs in the following proof?
Briefly justify your answer.
7= 7 – 1
2(7) + 3= 2(7 – 1) + 3
14 + 3= 2(6) + 3
17= 12 + 3
17 = 15
Math 20-2 Final Review
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Unit 2: Logic
Unit 3: Geometry
Vocabulary:
Corresponding angles: One interior angle and one exterior angle that are
non-adjacent and on the same side of a transversal.
Interior Angles: Any angles formed by a transversal and two parallel lines
that lie inside the parallel lines.
a
b
c
a, b, c, d are
interior angles.
d
Exterior Angles: Any angles formed by a transversal and two parallel lines
that lie outside the parallel lines.
e
f
e, f, g, h are
exterior angles.
g h
Supplementary angles: Two angles that add to 180˚.
Alternate interior angles: Two non-adjacent interior angles on opposite sides
of a transversal.
Math 20-2 Final Review
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Unit 3: Geometry
Alternate Exterior angles: Two exterior angles formed between two lines and
a transversal, on opposite sides of the transversal.
Non-adjacent interior angles: The two angles of a triangle that do not have
the same vertex as an exterior angle.
A
B
C
D
∠ A and ∠ B are non-adjacent interior angles to exterior ∠ACD
Convex polygon: A polygon in which each interior angle measures less than
180˚.
Key Ideas:
If a transversal intersects two lines such that




The corresponding angles are equal or
The alternate interior angles are equal or
The alternate exterior angles are equal or
The interior angles on the same side of the transversal are
supplementary,
Then the lines are parallel.
The sum of the measures of the interior angles of a convex polygon with n
sides can be expressed as: 180˚(n – 2).
The measure of each interior angle of a regular polygon is:
Math 20-2 Final Review
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180°(𝑛−2)
𝑛
.
Unit 3: Geometry
The sum of the measures of the exterior angles of any convex polygon is
360˚
If three pairs of corresponding sides are equal, then the triangles are
congruent. This is known as side-side-side congruence, or SSS.
If two pairs of corresponding sides and the contained angles are equal, then
the triangles are congruent. This is known as the side-angle-side
congruence or SAS
If two pairs of corresponding angles and the contained sides are equal, then
the triangles are congruent. This is known as the angle-side-angle
congruence or ASA
The following Acronym can be helpful in remembering the ratios:
SOH
sin =
opposite
hypotenuse
CAH
cos =
TOA
adjacent
hypotenuse
tan =
opposite
adjacent
The sine law can be used to determine unknown side lengths or angle
measures in acute triangles.
You can use the sine law,
sin 𝐴
𝑎
=
sin 𝐵
𝑏
=
sin 𝐶
𝑐
, to solve a problem modeled by an
acute triangle when you know:


Two sides and the angle opposite a known side
Two angles and any side
If you know the measures of two angles in a triangle, you can determine the
third angle because the angles must add to 180˚.
Some oblique triangle cannot be solved using the Sine Law. Therefore when
you are not given a side and its angle, you can use the Cosine Law,
c² = a² + b² - 2abCosC.
Math 20-2 Final Review
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Unit 3: Geometry
Practice Questions:
Multiple Choice
Identify the choice that best completes the statement or answers the
question.
____ 1. Which angle property proves DAB = 120°?
a.
b.
c.
d.
alternate exterior angles
corresponding angles
vertically opposite angles
alternate interior angles
____ 2. Which are the correct measures for DCE and CAB?
a.
b.
c.
d.
DCE
DCE
DCE
DCE
=
=
=
=
31°,
47°,
13°,
37°,
CAB
CAB
CAB
CAB
=
=
=
=
134°
109°
143°
119°
____ 3. Which are the correct measures for NOK and JON?
a.
b.
c.
d.
NOK
NOK
NOK
NOK
=
=
=
=
35°,
35°,
38°,
28°,
JON
JON
JON
JON
=
=
=
=
82°
36°
35°
63°
____ 4. Which angle is equal to D?
a.
b.
c.
d.
A
B
C
none of the above
Math 20-2 Final Review
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Unit 3: Geometry
____ 5. Determine the measure of  to the nearest degree.
a.
b.
c.
d.
40°
38°
36°
42°
____ 6. Determine the length of PQ to the nearest tenth of a centimetre.
a.
b.
c.
d.
8.8
8.5
9.1
9.4
cm
cm
cm
cm
____ 7. A kayak leaves Rankin Inlet, Nunavut, and heads due south for 4.5 km. At
the same time, a second kayak travels in a direction S75°E from the inlet for
3.2 km. In which direction, to the nearest degree, would the second kayak
have to travel to meet the first kayak?
a.
b.
c.
d.
S45°W
S45°W
S50°W
S40°W
____ 8. How long, to the nearest inch, is the right rafter in the roof shown?
a.
b.
c.
d.
33’0”
34’6”
33’6”
34’0”
Math 20-2 Final Review
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Unit 3: Geometry
Short Answer
9. Given QP || MR, determine the measure of MOQ.
10. Determine the value of x.
11. Determine the sum of the measures of the interior angles of this seven-sided
polygon.
Show your calculation.
12. In LMN, l = 10.0 cm, m = 13.2 cm, and M = 79°. Determine the measure
of L to the nearest degree.
13. Determine the measure of  to the nearest degree.
14. In GHI, g = 30.0 cm, i = 19.3 cm, and H = 53°. Determine the measure
of h to the nearest tenth of a centimetre.
Math 20-2 Final Review
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Unit 3: Geometry
15. Given z = 115°.
Determine the measure of y.
16. MO and LN are angle bisectors. What is the relationship, if any, between L
and O? Explain.
17. HJKL is a rectangle.
Prove: JM = NL
18. Determine the length of EF. Show your reasoning.
19. A radio tower is supported by two wires on opposite sides. On the ground,
the ends of the wire are 235 m apart. One wire makes a 75° angle with the
ground. The other makes a 55° angle with the ground.
Draw a diagram of the situation. Then, determine the length of each wire to
the nearest metre. Show your work.
Math 20-2 Final Review
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Unit 3: Geometry
Unit 4: Radicals
Vocabulary:
Extraneous Root: A root that does not satisfy the initial conditions that were
introduced while solving an equation. Root is another word for solution.
Key Ideas:
A radical is in simplest form when the exponent of the radicand is less than
3
the index of the radical. For example, 12√3 and 13 √4 are in simplest form,
while 12√4 is not.
If you express an answer as a radical, the answer will be exact. If you write
a radical in decimal form, the answer will be an approximation, except when
the radicand is a perfect square. For example, √12 expressed as 2√3
remains an exact value, while √12 expressed as 3.464… is an approximation.
Both √9 and 3 are exact values.
The product of two square roots is equal to the square root of the product.
√3 ∙ √2 = √3 ∙ 2 = √6
The product of two mixed radicals is equal to the product of the rational
numbers times the product of the radicals.
3√2 ∙ 5√7 = 15√14
The quotient of two square roots is equal to the square root of the quotient:
√6
√2
= √3
The quotient of two mixed radicals is equal to the product of the quotient of
the coefficients and the quotient of the radicals:
15√14
5√7
Math 20-2 Final Review
= 3√2
19
Unit 4: Radicals
Practice Questions:
Multiple Choice
Identify the choice that best completes the statement or answers the
question.
____ 1. Which of these equations are true?
I.
II.
III.
a.
b.
c.
d.
II only
I and II
I, II, and III
I only
____ 2. Which is the simplest form of –7
–2
–6
?
a. –47
b. –161
c. –15
d.
____ 3. Which expression is the rationalized form of
?
a.
b.
c.
d.
Short Answer
4. Convert –
5. Express
+
–
into mixed radical form. Then simplify.
in mixed radical form and entire form.
Math 20-2 Final Review
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Unit 4: Radicals
6. A park has a width of
the park.
m and a length of
m. Determine the area of
7. State any restrictions on the variable, then divide.
8. Highway engineers design on-ramps and off-ramps to be safe and efficient.
The relationship between the maximum speed at which a car can travel
around a curve without skidding is S =
, where S is the speed in
kilometres per hour and R is the radius in metres of an unbanked curve in
metres. What is the maximum speed at which a car can travel on a ramp
with a radius of 200 m, to the nearest tenth of a metre?
9. Buildings in snowy areas often have steeper roofs than buildings in drier
areas. This steepness, or “pitch,” is expressed as the height of a roof divided
by its width. Determine the pitch, in lowest mixed radical form, for a building
whose roof is
m high and
m wide. Show your work.
10. Simplify
Math 20-2 Final Review
. Explain each step.
21
Unit 4: Radicals
Unit 5: Statistics
Vocabulary:
Mean: What is commonly called the average, calculated by adding up all the
numbers and dividing by how many numbers there are.
The mean of a sample is 𝑥̅ . The mean of a population is 𝜇.
Median: The middle value, provided the data has been organized in
ascending order. The mean of the two middle values if there is an even
number of values.
Mode: The most frequent value.
There can be more than one mode, if more than one value ties for the most
frequent. If all values appear with the same frequency, there is no mode.
Range: The difference between the top and bottom numbers in a data set.
Outliers: A value in a data set that is far from the other values in the data
set.
Line Plot: A ‘graph’ that shows each number in the data set as a point above
a number line.
Dispersion: A measure that varies by the spread among the data in a set;
dispersion has a value of zero if all the data in a set is identical, and it
increases in value as the data becomes more spread out.
Frequency: is the number of times the data value occurs in a set.
For example, if four students have a score of 80 in mathematics, and then
the score of 80 is said to have a frequency of 4. The frequency of a data
value is often represented by f.
Frequency Table: a table constructed by arranging collected data values in
ascending order of magnitude with their corresponding frequencies
Histogram: a graph of a frequency distribution, in which equal intervals of
values are marked on a horizontal axis and the frequencies associated with
these intervals are indicated by the areas of the rectangles drawn for these
intervals.
Math 20-2 Final Review
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Unit 5: Statistics
Standard Deviation: A measure of the dispersion or scatter of data values in
relation to the mean; a low standard deviation indicates that most data
values are close to the mean, and a high standard deviation indicates that
most data values are scattered farther from the mean.
Normal Curve: a symmetrical curve that represents the normal distribution;
also called the bell curve.





50% of the data is above the mean
68.26% of the data is within one standard deviation of the mean
95.44% of the data is within two standard deviations of the mean
99.74% of the data is within three standard deviations of the mean
Total area under the curve is 1 or 100%
Normal Distribution: data that, when graphed as a histogram or a frequency
polygon, results in a unimodal symmetric distribution about the mean.
Z-Score: is a standardized value that indicates the number of standard
deviations of a data value above or below the mean.
Math 20-2 Final Review
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Unit 5: Statistics
Key Ideas:



A confidence interval is expressed as the survey or poll result, plus or
minus the margin of error.
The margin of error increases as the confidence level increases (with a
constant sample size). The sample size that is needed also increases
as the confidence level increases (with a constant margin of error).
The sample size affects the margin of error. A larger sample results in
a smaller margin of error. A larger sample results in a smaller margin
of error, assuming that the same confidence level is required.
Practice Questions:
Multiple Choice
Identify the choice that best completes the statement or answers the
question.
____ 1. Determine the z-score for the given value.
µ = 510,  = 93, x = 412
a. 0.95
b. –0.95
c. 1.05
d. –1.05
____ 2. Determine the percent of data between the following z-scores:
z = 0.40 and z = 1.80.
a. 7.72%
b. 22.66%
c. 15.44%
d. 30.87%
Short Answer
3. Joel researched the average daily temperature in his town.
Average Daily Temperature in Lloydminster, SK
Month
Jan. Feb. Mar. Apr. May Jun.
average daily –10.0 –17.5 –5.0 3.7 10.7 14.3
temperature (°C)
Jul.
20.1
Aug.
14.0
Sep.
9.8
Oct.
4.8
Nov. Dec.
–5.8 –14.8
Determine the median of the data.
Math 20-2 Final Review
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Unit 5: Statistics
4. A teacher is analyzing the class results for a computer science test. The marks
are normally distributed with a mean (µ) of 77.4 and a standard deviation ()
of 4.2.
Determine Sina’s mark if she scored µ + 2.5.
5. In a recent survey of high school students, 43% of those surveyed agreed that
the lunch period should be extended. The survey is considered accurate to
within 4.6 percent points, 19 times out of 20.
If a high school has 1500 students, state the range of the number of students
who would agree with the survey.
6. Leon keeps track of the amount he spends, in dollars, on weekly lunches
during one semester:
25
19
36
19
17
10
24
33
24
28
25
31
28
26
29
26
18
32
a) Determine the range, mean, and standard deviation, correct to two decimal
places.
b) Remove the greatest and the least weekly amounts. Then determine the
range, mean, and standard deviation for the remaining amounts. What effect
does removing the greatest and the least amounts have on the three values?
7. Yumi always waits until her gas tank is nearly empty before refuelling. She
keeps track of the distance she drives on each tank of gas. The distance varies
depending on the weather and the amount she drives on the highway. The
distance has a mean of 520 km and a standard deviation of 14 km.
a) What percent of the time does she drive between 534 km and 562 km on
a tank of gas?
b) Between what two symmetric values will she drive 95% of the time?
Math 20-2 Final Review
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Unit 5: Statistics
8. Jackson raises Siberian husky sled dogs at his kennel. He knows, from the
data he has collected over the years, that the masses of adult male dogs are
normally distributed, with a mean of 23.6 kg and a standard deviation of 1.8
kg. Jackson has 48 puppies this year. How many of them could he expect to
have a mass greater than 20 kg when they grow up?
9. A hardware manufacturer produces bolts that have an average length of
1.22 in., with a standard deviation of 0.02 in. To be sold, all bolts must have
a length between 1.20 in. and 1.25 in. What percent, to the nearest whole
number, of the total production can be sold?
10. Use confidence intervals to interpret the following statement and apply the
result to a graduating class of 1400 students.
In a recent survey, 72% of post-secondary graduates indicated that they
expected to earn at least $6000/month by the time they were ready to
retire. The survey is considered accurate within ±5.2%, 19 times in 20.
11. Two different market research companies conducted a survey on the same
issue. Company A used a 90% confidence level and company B used a 95%
confidence level.
a) If both companies used the same sample size, what does this imply about
the margin of error for each survey?
b) If both companies used the same margin of error of ±3.5%, what does
this imply about the sample size for each survey?
Math 20-2 Final Review
26
Unit 5: Statistics
Unit 6: Proportional Reasoning
Formulas:
1
𝐴𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒 = 2 𝑏ℎ
𝐴𝑐𝑖𝑟𝑐𝑙𝑒 = 𝜋𝑟²
𝐴𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙𝑜𝑔𝑟𝑎𝑚 = 𝑏ℎ
h
b
1
𝐴𝑡𝑟𝑎𝑝𝑒𝑧𝑜𝑖𝑑 = 2 ℎ(𝑎 + 𝑏)
a
h
b
If the area of a similar 2-D shape and the area of the original shape are
known, then the scale factor, k, can be determined using the formula:
𝑘2 =
𝑎𝑟𝑒𝑎 𝑜𝑓 𝑠𝑖𝑚𝑖𝑙𝑎𝑟 2 − 𝐷 𝑠ℎ𝑎𝑝𝑒
𝑎𝑟𝑒𝑎 𝑜𝑓 𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑠ℎ𝑎𝑝𝑒
1
𝑆𝐴𝑟𝑖𝑔ℎ𝑡 𝑡𝑟𝑖𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑝𝑟𝑖𝑠𝑚 = 𝑏ℎ + 𝑙(𝑎 + 𝑏 + 𝑐)
𝑉𝑟𝑖𝑔ℎ𝑡 𝑡𝑟𝑖𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑝𝑟𝑖𝑠𝑚 = 2 𝑏ℎ𝑙
𝑆𝐴𝑟𝑖𝑔ℎ𝑡 𝑐𝑦𝑙𝑖𝑛𝑑𝑒𝑟 = 2𝜋𝑟 2 + 2𝜋𝑟ℎ
𝑉𝑟𝑖𝑔ℎ𝑡 𝑐𝑦𝑙𝑖𝑛𝑑𝑒𝑟 = 𝜋𝑟 2 ℎ
1
𝑆𝐴𝑟𝑖𝑔ℎ𝑡 𝑝𝑦𝑟𝑎𝑚𝑖𝑑 = 𝑙² + 2𝑙𝑠
𝑉𝑟𝑖𝑔ℎ𝑡 𝑝𝑦𝑟𝑎𝑚𝑖𝑑 = 3 𝑙²ℎ
𝑆𝐴𝑟𝑖𝑔ℎ𝑡 𝑐𝑜𝑛𝑒 = 𝜋𝑟² + 𝜋𝑟𝑠
𝑉𝑟𝑖𝑔ℎ𝑡 𝑐𝑜𝑛𝑒 = 3 𝜋𝑟²ℎ
1
4
𝑆𝐴𝑠𝑝ℎ𝑒𝑟𝑒 = 4𝜋𝑟²
Math 20-2 Final Review
𝑉𝑠𝑝ℎ𝑒𝑟𝑒 = 3 𝜋𝑟³
27
Unit 6: Proportional Reasoning
Key Ideas:
𝐷𝑖𝑎𝑔𝑟𝑎𝑚 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑚𝑒𝑛𝑡

To determine a scale factor:



Scale factor between 0 and 1 is a reduction
Scale factor greater than 1 is an enlargement
Two 3-D objects that are similar have dimensions that are
𝐴𝑐𝑡𝑢𝑎𝑙 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑚𝑒𝑛𝑡
proportional.

The scale factor is the ratio of a linear measurement of an object to
the corresponding linear measurement in a similar object, where both
measurements are expressed using the same units.

To create a scale model or diagram, determine an appropriate scale to
use based on the dimensions of the original shape and the size of the
model or diagram that is required
Practice Questions:
Multiple Choice
Identify the choice that best completes the statement or answers the
question.
____ 1. It costs $0.3172/lb to ship freight by barge along the Pacific coast.
Which equation determines the cost, C, in dollars, to ship 2740 kg
of building supplies from Vancouver to Prince Rupert?
a.
b.
c.
d.
Math 20-2 Final Review
28
Unit 6: Proportional Reasoning
____ 2. The distance between two towns on a map is 16.5 cm. The map
was made using a scale of 5 cm to 100 km. What is the actual
distance between the two towns?
a.
b.
c.
d.
135
825
330
165
km
km
km
km
____ 3. A Ferris wheel is 57.85 m tall and has a diameter of 54.4 m.
What are the dimensions of a scale model built using a scale
of 1 : 95?
a.
b.
c.
d.
height
height
height
height
64.22
57.26
60.89
60.89
cm,
cm,
cm,
cm,
diameter
diameter
diameter
diameter
60.44
53.42
64.22
57.26
cm
cm
cm
cm
____ 4. Cylinder A has a radius of 5 mm and a height of 30 mm. Cylinder B has a
radius of 20 mm and a height of 120 mm. These two cylinders are similar.
By what factor is the volume of cylinder B greater than the volume of
cylinder A?
a.
b.
c.
d.
64
8
16
32
Short Answer
5. A reindeer can run 133.25 km in 2.5 h. A grizzly bear can run 12.5 km in 15
min. Determine the speed of each animal in kilometres per hour. Which
animal can faster?
6. It takes 5 h 17 min to decorate 200 cupcakes. How many minutes will it take
to decorate three dozen cupcakes?
Math 20-2 Final Review
29
Unit 6: Proportional Reasoning
7. Today, gold is worth $1200.60/oz (1 oz = 28.3495 g). What is the value of
0.9 g of gold?
8. Leo has a microscope with a lens that magnifies by a factor of 80.
He was able to capture the image of a slide containing human skin
cells. In the image, the cell was about 5.6 mm long. Determine the
length of the actual human skin cell, to nearest hundredth of a
millimetre.
9. The radius of a circle with an area of 8 cm2 will be enlarged by a scale factor
of 4.
Determine the area of the enlarged circle.
10. Museum curators are building scale models of antique furniture for a
children's activity area.
A chair is 91 cm tall, 56 cm wide, and 58 cm long. They would like the scale
model to be 13 cm tall.
What scale factor should they use?
11. The following table shows the attendance figures for the Calgary Stampede
over several years. During which period was attendance decreasing at the
greatest rate? Justify your answer.
Year
1990
1995
1999
2004
2008
2010
Math 20-2 Final Review
Attendance
1 208 371
1 101 551
1 113 017
1 221 182
1 236 351
1 145 394
30
Unit 6: Proportional Reasoning
12. Monroe and Connie drove from Winnipeg to Lethbridge for a music
festival. They took turns driving, so they only needed to stop for
gas or food. They drove the 1202 km distance in 15 h 34 min. They
used 127.8 L of fuel, which cost $142.54.
a) Determine their average speed to the nearest tenth of a kilometre per
hour.
b) Determine their average fuel consumption per 100 km.
c) What was the average cost of a litre of gas?
13. A cook has a set of four mixing bowls with lids. The bowls stack inside
each other and are similar to each other. The surface areas of the two
largest bowls are 2300 cm2 and 1100 cm2. The scale factor is the same
from each bowl to the next smaller bowl. How would you find the scale
factor for the diameters of the bowls?
Math 20-2 Final Review
31
Unit 6: Proportional Reasoning