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22.Water and coffee are two of the most common drinks throughout the world. The table below ranks 12 countries
according to their per capita coffee consumption and also provides the relative ranking of those countries in
terms of per capita bottled water consumption.
a . Will the rank correlation of this data be positive or negative? Explain how
you can determine this by just looking at the scatterplot.
b. Use the formula
= _________
to calculate the rank correlation for this set of data.
23. The scatterplot matrix below indicates the mean annual amounts of three different pollutants
(measured in micrograms per cubic meter) in 11 selected European cities. The pollutants
considered are suspended particulates, sulfur dioxide, and nitrogen dioxide. Each of the points
corresponds to one city.
a. Which pair of variables has the strongest association? Explain your reasoning.
b. The circled point in each scatterplot corresponds to Athens, Greece. Use the scatterplot matrix to estimate
the amount of each type of pollutant in Athens, Greece.
Suspended Particulates: _______________
Sulfur Dioxide: _______________
Nitrogen Dioxide: ________________
c. On the (suspended particulates, nitrogen dioxide) scatterplot, the point with coordinates
(77, 248) is an outlier. Identify what type of outlier it is. Explain your reasoning.
d. Give possible coordinates for a point that would be an outlier for both variables for the
(suspended particulates, sulfur dioxide) data.
24. The scatterplot below shows the relationship between engine size (in liters) and highway miles per gallon for
many small cars.
a. Describe the shape of the scatterplot.
b. Add a point to the scatterplot that would be an outlier only when the engine size and mpg are considered
jointly. Label your point C. Explain why it is an outlier only when both variables are considered together.
25. The scatterplot below indicates the birth rates (per 1,000 people) and the death rates (per 1,000 people) for 11
different states in the western United States. The regression line is shown on the plot and has equation y = –
0.334x + 12.5.
a. Which of the following is the best estimate of the correlation between birth rate and death rate for these 11
states? Explain your reasoning.
r = -0.82
r = -0.32
r = 0.32
r = 0.82
b. The circled point is the point for Washington. It has coordinates (13, 7.4). On the scatterplot, draw in the
line segment representing the residual for that point.
c. Use the equation for the regression line to calculate the residual for Washington, and then
explain what it tells you.
d. On the scatterplot, circle the point that is an outlier. Then, describe how the slope of the
regression line and the correlation would change if that point were deleted from the data set.
Slope of regression line:
Correlation:
26. The eighth-graders at Claremont Middle School determined their best times (in minutes) for
running both a quarter mile Q and one mile M. The regression equation for the line of best fit is M = 5.25Q +
0.18.
a. Explain the meaning of the slope of the regression line.
b. Circle the vertex-edge graph that you think best illustrates the relationship between the two
running times. Explain your reasoning.
c. The mean of the quarter-mile running times was 2.8 minutes. Find the mean running time for the one-mile
run.
d. Suppose that the times are changed from minutes to seconds. How will the correlation for this new set of
data compare to the correlation for the original set of data? Explain your reasoning.
27. Consider the following summary of a story reported on the radio.
a. What are the explanatory and response variables identified in the news report?
Explanatory variable: _____________________
Response variable: _______________________
b. Do you agree or disagree with the statement “that encouraging people to brush more could help prevent
obesity”? Explain your reasoning.
c. Identify the lurking variable mentioned in the report.
28. Jenny needs to build a box that has a square base and a volume of 900 cm . The height of the box is related to
the length of one side of the base by the function h(s) =
.
a. What value of y satisfies the equation y = h(10). What does this tell you about the box?
b. What value(s) of s satisfy the equation 100 = h(s)? Do all of your solutions make sense in this context?
Explain your reasoning.
c. Describe the theoretical domain and range for h(s).
d. What are reasonable practical domain and range for h(s)?
29. a. Complete the table of values below so that y is not a function of x. Explain.
Explanation:
b. True or False: If x is a function of y, then y is a function of x. Explain your reasoning.
30. Write a rule for a quadratic function with a graph that has x-intercepts (–2, 0) and (8, 0) and
y-intercept (0, 8).
31. Write each product in equivalent ax + bx + c form.
a. (x - 8)(x - 4)
b. (3x + 2)(x + 5)
c. (x + 9)
d. (x + 4)(x - 4)
32. Find an equivalent factored form for each quadratic expression.
a. x - 10x + 21
b. x - 100
c. x + x - 56
d. x + 8x + 16
e. 1 - 2x + x
f. 14x + 18x
g. 2x + 10x + 8
33. Solve each quadratic equation. Use factoring at least once and use the quadratic formula at least once. Show
your work.
a. x + 9x + 18 = 0
b. 3x - 17 = -8
c. x - 3x + 3 = 31
d. 2x - 3x - 1 = 2
e. 5x = 25x
34. a. Write a quadratic equation in the form ax + bx + c = 0 that has solutions of x = -5 and x = 8.
b. Is the equation that you wrote in Part a the only quadratic equation of that form that has the
indicated solutions? Explain your reasoning.
35. The Hillsdale Rock the Vote committee is organizing a community concert and voter registration drive. The
cost for running the concert is $2,470. They need to decide what the minimum admission charge for the
concert should be so that they do not lose any money.
a. If they charge $5 admission, how many people would need to attend the concert in order for the income
from admission charges to equal the cost of running the concert?
b. Write a rule that indicates how the number of people that would need to attend the concert N depends on
the admission charge c.
c. The formula A = 640 - 40c indicates the number of people A likely to attend the concert if the admission
charge is c dollars. Will they make money or lose money if they charge $4? Explain your reasoning or show
your work.
d. Write and solve an equation that will identify the minimum charge in order for the concert to attract enough
people to cover its costs.
36. Consider this system of equations:
y = 3x + 25 and y = 2x + 14x - 15
a. Use graphs or tables of values to solve this system of equations. Explain your reasoning and
include sketches of graphs or tables of values showing the solution.
b. Solve the system of equations by reasoning with the symbolic form
37. Solve this equation by reasoning with the symbols.
=x+1
38. Explain how it is possible for a system of equations involving one linear equation and one quadratic equation
to have no real number solutions.
39. Without using your calculator, determine if each of the following statements is true or false. Explain your
reasoning for each part.
a. log 10,000 = 4
b. log 0.01 =
c. There is no value of x such that 10 = 54.
40. Without using your calculator, determine which of the following statements is true. Explain how you decided
the statement as true.
Statement I
900
Statement II
2
log 925
3
Statement III 3
log 925
4
Statement IV
log 925
10
9
log 925
1,000
41. Solve each equation without using tables or graphs. Show your work.
a. 10 = 71
b. 10 + 1 = 4,170
42. Recall that a sound with intensity 10 watts/cm has a decibel rating of 10x + 120. The sound
intensity of a popping balloon is 5,011 watts/cm . What is the decibel rating for a popping balloon?
43. The amount of light that is able to pass through the water in a lake depends on the clarity of the water.
Suppose that in one lake, the function rule for light intensity (measured in lux) at a depth of d meters is I(d) =
60,000
.
a. What is the light intensity at a depth of 1 meter?
b. Determine the depth at which the light intensity will be 3,000 lux. Show your work.
44. Consider the line that contains the origin and the point with coordinates (3, 4).
a. Sketch this line and find the equation of the line.
b. Label the angle formed by the line and the positive x-axis . Without using your calculator,
express the following in ratio form.
tan
=
cos
=
sin
=
45. Triangle ABC is shown below.
a. Compute the length of
AC =
.
b. Without using your calculator, find the following. Express your answers in ratio form.
sin A =
cos A =
tan A =
46. Usually, it is not possible to measure the heights of very tall structures, like towers or flagpoles, with a tape
measure. Surveyors use right triangle trigonometry, measuring lengths that are accessible with a tape measure
(or other tool) and angles between two lines of sight with a transit.
a. A flagpole and transit are pictured below. Sketch and label the vertices of a right triangle with an angle that
could be measured by the transit, a side that could be measured with a tape measure, and a second side that is
(nearly) the height of the flagpole. From your sketch, name the angle and sides that fit the above description.
Angle measured by transit:______________
Side measured by tape measure:______________
Side that is (nearly) flagpole height:______________
b. Rita uses the transit, which is 1.6 m high and 12 m from the flagpole (on level ground), to sight to the top
of the flagpole. The angle of elevation to the top of the flagpole measures 58. Label these measures on the
sketch below.
c. Find the height of the flagpole to the nearest tenth of a meter. Explain your reasoning.
Height of flagpole =______________
47. James needs to attach a stabilizing wire to a tall tower. The wire is 200 feet long and should be attached to the
tower at a height of 100 feet. Assume that the ground around the tower is level and that the entire length of the
wire is used.
a. Draw a sketch of this situation and label any known lengths.
b. Determine the angle that the wire will make with the ground.
c. Find the distance from the tower to the point where the wire is attached to the ground.
48. Melissa and Rosa are golfing on a beautiful summer day. The ninth hole is 380 yards. Melissa
hooked (hit to the left of the correct direction) her drive on hole #9 as sketched below.
How far is Melissa’s ball from the hole? Explain or show your work.
49. A triangular region has sides measuring 25, 35, and 15 feet. Find the measure of the angle opposite the 35
foot long side.
50. Suppose that you know the measures of all three angles in
and that NP = m and MN = p. Determine
whether or not each of the following would give the correct length for
.
a. MP =
b. MP =
c. MP =
d. MP =
51. To know how much paint is needed for a barn, a farmer is estimating the total surface area of the barn. One
part of the surface is triangular as sketched below.
a. The darkened sides in the figure are the edges of the roof. This trim will be painted white. Find the length
of each of these two sides of the triangle. Explain or show your work.
b. The triangular surface needs to be painted red. Find the area of the triangle. Explain or show your work.
52. Suppose that you are randomly selecting a person from a town with a population of 25,000.
Consider the following events:
• Person is a female.
• Person has a driver’s license.
• Person owns a car.
• Person is younger than 12 years old.
• Person is older than 21 years old.
• Person has read Hamlet.
For each part, identify events A and B from the list above that will make the equation true. Then explain your
reasoning. You may use events more than once.
a. P(A or B) = P(A) + P(B)
Event A ____________________
Event B ________________________
b. P(A and B) = P(A) • P(B)
Event A ____________________
Event B ________________________
c. P(A and B) = 0
Event A ____________________
Event B ________________________
d. P(A) P(A | B)
Event A ____________________
Event B ________________________
53. In Colorado, approximately 20% or 1 in 5 people over the age of six belong to a gym.
(Source: USA Today)
a. Suppose that you randomly select two people who are over
the age of six and live in Colorado. Draw an area model and
use it to determine the probability that only one of the two
people belongs to a gym.
Area Model
b. Suppose you randomly select three people who are over the
age of six and live in Colorado. Find the probability that all
three of them belong to a gym.
c. Suppose Austin randomly selects one person over the age of six who lives in Colorado. Since about half of
the people over the age of six in Colorado are male, he determines the probability of getting a male who
belongs to a gym by doing the following calculation:
P(male and belongs to a gym) =
=
Is Austin’s calculation correct? Explain your reasoning.
54. The table below indicates the number of physicians in Minnesota in 2005 by age and gender.
Suppose that you randomly choose one of these physicians to interview. Find each of the following
probabilities. (Source: Minnesota Physicians Facts and Data 2005)
a. P(female)
b. P(under 65)
c. P(female and under 65)
d. P(female or under 65)
e. P(female | under 65)
f. P(under 65 | female)
g. Are gender and age of physicians in Minnesota in 2005 independent? Support your reasoning using
information from the table.
55. Approximately 12% of the U.S. population wears contact lenses.
a. If you randomly select 200 people, how many would you expect to wear contact lenses?
b. If you randomly select 70 people how many would you expect to wear contact lenses?
56. The Taylor Art Association is planning to have a fund-raiser each month at their monthly art show. People
will pay to spin the spinner below and will win a gift certificate with the indicated value.
a. Complete the probability distribution table for the outcome of one spin .
b. If the Art Association charges $20 for one spin, should they expect to make money, lose money, or break
even over the long run? Show your work or explain your reasoning.
c. What is the fair price to charge for a spin of this spinner? Show your work.
d. Over the course of the year, they expect to have 500 people pay to spin the spinner. What should they
charge for a spin if they want to have at least $2,000 to provide scholarships to Art Camp? Explain your
reasoning.
57. The prizes in a raffle are five $25 gift certificates to a local restaurant, three weekend getaways each worth
$650, and the grand prize of a trip for two to Washington, D.C. worth $2,375. Exactly 3,000 raffle tickets will
be sold.
a. If you buy one raffle ticket, what is the probability that you will win a weekend getaway?
b. What is the fair price to charge for one raffle ticket? Show your work.
ANSWERS
22. ANS:
a. The correlation will be negative since the points are clustered about a line with negative slope.
b.
= 0.74
PTS: 1
REF: Lesson 4-1
OBJ: 4-1.3 Identify types of association (positive and negative, strong and weak, perfect and none, linear
and nonlinear).
NAT: 1 | 2
23. ANS:
a. The variables suspended particulates and sulfur dioxide have the strongest association. The points in the
scatterplots for these variables cluster more closely about a line than do the points in the other scatterplots.
b. Suspended Particulates: approximately 175–180 micrograms per cubic meter
Sulfur Dioxide: approximately 35 micrograms per cubic meter
Nitrogen Dioxide: approximately 65 micrograms per cubic meter
c. This point is an outlier for nitrogen dioxide only. The 248 micrograms per cubic meter is much greater than
any other nitrogen dioxide value, but the 77 micrograms per cubic meter is in the middle of the range of
amounts for suspended particulates.
d. Responses will vary. Since the highest amount of suspended particulates is slightly less than
200 micrograms per cubic meter and the highest amount of sulfur dioxide is approximately
40 micrograms per cubic meter, a point such as (400, 100) would be an outlier for both variables.
PTS: 1
REF: Lesson 4-1
OBJ: 4-1.3 Identify types of association (positive and negative, strong and weak, perfect and none, linear
and nonlinear). | 4-1.3 Identify types of association (positive and negative, strong and weak, perfect and none,
linear and nonlinear). | 4-1.4 Identify clusters and different types of outliers. | 4-1.5 Read and interpret a
scatterplot matrix.
NAT: 1 | 2 | 7 | 8
24. ANS:
a. The plot is curved, and it fans out to the right. There is more variability in the mpg for large
engines than for small engines.
b. Responses will vary. Points with coordinates such as (4, 40) or (1.5, 15) would be considered outliers when
engine size and mpg are considered jointly. If these points are considered by engine size and mpg separately,
they are not out of the ordinary.
PTS: 1
REF: Lesson 4-1
OBJ: 4-1.2 Describe shapes of clouds of points on scatterplots (linear, curved, vary in strength). | 4-1.4
Identify clusters and different types of outliers.
NAT: 1 | 2 | 7 | 8
25. ANS:
a. r = -0.82
Since the regression line has negative slope, the value of r must be negative. Since the points
cluster fairly close to the line, r should be fairly close to -1.
b. The line segment should be a vertical segment from the point to the line.
c. Residual = 7.4 - [-0.334(13) + 12.5] = -0.758. The residual indicates that the death
rate is 0.758 smaller than what would be predicted for a birth rate of 13.
d. The point in the bottom right-hand corner should be circled. When this outlier is
eliminated, the regression line will get steeper, so the slope will get more negative (it will decrease).
The correlation becomes less strong, so the correlation will be closer to zero but will still be negative. The
absolute value of the correlation will decrease.
PTS: 1
REF: Lesson 4-1 | Lesson 4-2
OBJ: 4-1.3 Identify types of association (positive and negative, strong and weak, perfect and none, linear
and nonlinear). | 4-2.1 Understand that a linear model is appropriate when the points form an elliptical cloud. |
4-2.2 Compute errors in prediction and residuals and locate them on the plot. | 4-2.5 Explore the effect of
outliers and influential points on the regression equation.
NAT: 1 | 2
26. ANS:
a. The slope of 5.25 means that if one student has a quarter-mile time that is 1 minute more than another
student, we would expect the 1-mile time to be 5.25 minutes greater.
b. Students should choose the second graph since a change in the time for running a quarter mile does not
cause a change in the time for running a mile. One obvious lurking variable is the overall fitness or athletic
ability of the student.
c, Since
must be on the regression line, the mean running time for the 1-mile run is
5.25(2.8) + 0.18 = 14.88 minutes.
d. The correlation will be the same. The position of the points in relation to each other is not
changed by this transformation, and so the correlation will not change.
PTS: 1
REF: Lesson 4-3
OBJ: 4-3.1 Review and synthesize the major objectives of the unit.
NAT: 1 | 2
27. ANS:
a. Explanatory variable: how often someone brushes his/her teeth
Response variable: weight
b. It is probably not the case that people will lose weight if they brush more often. This is not likely to be a
cause-and-effect relationship.
c. How much people care about their appearance and health
PTS: 1
REF: Lesson 4-2
OBJ: 4-2.11 Identify possible explanations (cause-and effect, lurking variable) for an association and
illustrate with a directed graph. | 4-2.12 Identify the explanatory variable and the response variable.
NAT: 1 | 2 | 7 | 8 | 9
28. ANS:
a. h(10) = 9. This indicates that if the sides of the base of the box are 10 cm long then the height of the box
will need to be 9 cm.
b. s = 3 and s = -3 satisfy the equation 100 = h(s). However, only s = 3 makes sense in this
situation since the length of the edge of a box cannot be -3 cm.
c. The theoretical domain is all nonzero numbers. The theoretical range is all positive numbers.
d. Responses may vary. Look for answers that indicate that students thought about what dimensions would be
practical for building a box. It seems reasonable that the height should be at least 1 cm and perhaps not more
than 100 cm. This gives a practical domain of 3 cm s 30 cm and a practical range of 1 cm h(s) 100
cm.
PTS: 1
REF: Lesson 5-1
OBJ: 5-1.2 Use f(x) notation to represent functions and the common questions about functions that arise in
applied problems. | 5-1.3 Identify domain and range of functions.
NAT: 1 | 2 | 9 | 10
29. ANS:
a. Responses may vary. Students will need to repeat one of the x values and provide a different y value and
support their numbers with an explanation.
b. False. For example, the height of a thrown ball is a function of time, but time is not a function of the height
of the ball because it may be at the same height at two different times on the way up and on the way down.
PTS: 1
REF: Lesson 5-1
OBJ: 5-1.1 Distinguish relationships between variables that are functions from those that are not.
NAT: 1 | 2 | 7
30. ANS:
PTS: 1
REF: Lesson 5-1
OBJ: 5-1.4 Construct rules for quadratic functions based on given properties such as x-intercepts, yintercept, and maximum/minimum point.
NAT: 1 | 2
31. ANS:
a.
b.
c.
d.
PTS: 1
REF: Lesson 5-1
OBJ: 5-1.5 Write quadratic expressions in equivalent expanded or factored form.
NAT: 1 | 2
32. ANS:
a.
b.
c.
d.
e.
f.
g.
or
or
PTS: 1
REF: Lesson 5-1
OBJ: 5-1.5 Write quadratic expressions in equivalent expanded or factored form.
NAT: 1 | 2
33. ANS:
a.
or
b.
c.
or
or
d.
e.
or
or
PTS: 1
REF: Lesson 5-1
OBJ: 5-1.6 Solve quadratic equations by factoring, by applying the quadratic formula, or by a CAS.
NAT: 1 | 2 | 6
34. ANS:
a. Responses may vary.
for any nonzero value of k will have the
indicated solution.
b. No. The constant k could have any nonzero value. Students may provide an equation different than the one
they gave in Part a to justify their answer.
PTS: 1
REF: Lesson 5-1
OBJ: 5-1.4 Construct rules for quadratic functions based on given properties such as x-intercepts, yintercept, and maximum/minimum point.
NAT: 1 | 2
35. ANS:
a.
people
b.
c. 480 people will attend the concert (640
will lose money.
40(4) = 480). So, they will only make 480(4) = $1,920 and they
d.
The solutions to this equation are 6.5 and 9.5. Students can solve this using tables of values,
graphs, or symbolic reasoning. The minimum charge in order to break even is $6.50.
PTS: 1
REF: Lesson 5-2
OBJ: 5-2.1 Write an equation or inequality to represent a question about a "real-life" situation involving a
comparison between a linear function and either an inverse variation or quadratic function.
NAT: 1 | 2 | 9
36. ANS:
a. The solutions are (2.5, 32.5) and (-8, 1). Students should indicate that the graphs intersect at
the points indicated above. Using tables of values, students should indicate that when x = 2.5
and when x = -8, both functions have the same y values.
b.
or
Substituting these x values back into the equations gives the solutions (2.5, 32.5) and (-8, 1).
PTS: 1
REF: Lesson 1-3 | Lesson 5-1 | Lesson 5-2
OBJ: 1-3.2 Solve linear systems by graphing, substitution, and elimination methods. | 5-1.6 Solve quadratic
equations by factoring, by applying the quadratic formula, or by a CAS. | 5-2.3 Estimate solutions to
equations in the form mx + d = ax2 + bx + c using tables or graphs and solve algebraically.
NAT: 1 | 2 | 3 | 6
37. ANS:
20 =
0=
0 = (x + 5)(x 4)
x=
or x = 4
PTS: 1
REF: Lesson 5-2
OBJ: 5-2.2 Estimate solutions to equations in the form ax + b = k / x using tables or graphs and solve
algebraically.
NAT: 1 | 2 | 6
38. ANS:
If the graphs of the line and the quadratic function do not intersect, then the system of equations will not have
any solutions. The graph below is an example of such a situation.
PTS: 1
REF: Lesson 5-2
OBJ: 5-2.1 Write an equation or inequality to represent a question about a "real-life" situation involving a
comparison between a linear function and either an inverse variation or quadratic function.
NAT: 1 | 2 | 9
39. ANS:
a. True; 10 = 10,000.
b. False; 10
0.01.
c. True; 10 will always be positive.
PTS: 1
REF: Lesson 5-3
OBJ: 5-3.3 Use common logarithms to solve exponential equations, both in and out of context.
NAT: 1 | 2
40. ANS:
Statement II is correct since 925 is between 10 = 100 and 10 = 1,000.
PTS: 1
REF: Lesson 5-3
OBJ: 5-3.2 Be able to rewrite any real number as a power of 10 by finding common logarithms. | 5-3.3 Use
common logarithms to solve exponential equations, both in and out of context.
NAT: 1 | 2
41. ANS:
a. 10 = 71
x = log 71 1.85
b. x + 1 = log 4,170
x = log 4,170 1 2.62
PTS: 1
REF: Lesson 5-3
OBJ: 5-3.3 Use common logarithms to solve exponential equations, both in and out of context.
NAT: 1 | 2
42. ANS:
Solving 10 = 5,011 tells us that x = log 5,011 3.7. Thus, the decibel rating is 10(3.7) + 120 157 decibels.
PTS: 1
REF: Lesson 5-3
OBJ: 5-3.3 Use common logarithms to solve exponential equations, both in and out of context.
NAT: 1 | 2
43. ANS:
a. 15,002 lux
b. 60,000(10
10
) = 3,000
=0.05
= log 0.05
Thus, at a depth of approximately 2.16 m, the light intensity will be 3,000 lux.
PTS: 1
REF: Lesson 5-3
OBJ: 5-3.3 Use common logarithms to solve exponential equations, both in and out of context.
NAT: 1 | 2
44. ANS:
a.
x
b. tan
=
; cos
=
; sin
=
PTS: 1
REF: Lesson 1-4 | Lesson 7-1
OBJ: 1-4.1 Review and synthesize the major objectives of the unit. | 7-1.1 Determine values of the sine,
cosine, and tangent functions of an angle in standard position in a square coordinate plane.
NAT: 1 | 2 | 3
45. ANS:
a.
b. sin A =
AC =5
; cos A =
; tan A =
PTS: 1
REF: Lesson 7-1
OBJ: 7-1.2 Determine the sine, cosine, and tangent of an acute angle in a right triangle, and determine the
angle given one of those ratios. | 7-1.3 Solve problems involving indirect measurement that can be modeled as
parts of a right triangle.
NAT: 1 | 2 | 4
46. ANS:
a. Responses will vary if students label the triangle differently.
Angle measured by transit:
Side measured by tape measure:
Side that is (nearly) flagpole height:
b.
c. Referring to the figure in Part b, the height of the flagpole is AB + 1.6 and BC is 12 meters. tan 58° =
so AB = 12 tan 58°
19.2 meters. The height of the flagpole is AB + 1.6
,
19.2 + 1.6 = 20.8 meters.
PTS: 1
REF: Lesson 7-1
OBJ: 7-1.2 Determine the sine, cosine, and tangent of an acute angle in a right triangle, and determine the
angle given one of those ratios. | 7-1.3 Solve problems involving indirect measurement that can be modeled as
parts of a right triangle.
47. ANS:
a.
NAT: 1 | 2 | 4
b. sin = 0.5
sin (0.5) =
= 30°
c. Students can use either the Pythagorean Theorem or trigonometric ratios to determine that the wire will be
attached to the ground at approximately 173 feet from the tower.
PTS: 1
REF: Lesson 7-1
OBJ: 7-1.2 Determine the sine, cosine, and tangent of an acute angle in a right triangle, and determine the
angle given one of those ratios. | 7-1.3 Solve problems involving indirect measurement that can be modeled as
parts of a right triangle.
NAT: 1 | 2 | 4
48. ANS:
cos 16° ;
118 yards
PTS: 1
REF: Lesson 7-2
OBJ: 7-2.1 Determine measures of sides and angles of triangles using the Law of Sines and Law of Cosines.
NAT: 1 | 2 | 4
49. ANS:
35 = 25 + 15
2(25)(15) cos A
cos A =
m
= 120°
PTS: 1
REF: Lesson 7-2
OBJ: 7-2.1 Determine measures of sides and angles of triangles using the Law of Sines and Law of Cosines.
NAT: 1 | 2 | 4
50. ANS:
a. not correct
b. correct
c. not correct
d. correct
PTS: 1
REF: Lesson 7-2
OBJ: 7-2.1 Determine measures of sides and angles of triangles using the Law of Sines and Law of Cosines.
NAT: 1 | 2 | 4
51. ANS:
a. The angle at the top is 180°
2(36) or 108°. Using the Laws if Sines,
=
;
m.
b. First, find the height of the triangle: sin 36° =
0.5(10.5066)(sin 36°)(17)
52.7896
; so, h = (sin 36°)(10.5066). Thus, area =
53 m .
PTS: 1
REF: Lesson 7-1 | Lesson 7-2
OBJ: 7-1.3 Solve problems involving indirect measurement that can be modeled as parts of a right triangle. |
7-2.1 Determine measures of sides and angles of triangles using the Law of Sines and Law of Cosines.
NAT: 1 | 2 | 4
52. ANS:
a. If P(A or B) = P(A) + P(B), then A and B must be mutually exclusive events. There are two
possibilities for A and B.
A: Person has a driver’s license.
B: Person is younger than 12 years old.
or
A: Person is younger than 12 years old.
B: Person is older than 21 years old.
b. If P(A and B) = P(A) P(B), then A and B are independent events. Student responses may vary. You should
look for students demonstrating what it means for events to be independent. One possibility is:
A: Person is female.
B: Person has read Hamlet.
These events are likely to be independent since there is no reason to think that being female would make one
more or less likely to have read Hamlet.
c. If P(A and B) = 0, then A and B must be mutually exclusive events. (See the answer to Part a for two
possible correct answers.)
d. Responses may vary. If A is “person has driver’s license” and B is “person is older than 21,” we would
expect P(A) P(A | B). This is the case because a person over 21 is more likely to have a driver’s license than
a person randomly selected from the entire population.
PTS: 1
REF: Lesson 8-1
OBJ: 8-1.2 Use the Multiplication Rule to find the probability that two independent events both occur. | 81.3 Find conditional probabilities and determine if two events are independent. | 8-1.4 Use the Multiplication
Rule to find the probability that two events both occur when the events are not independent.
NAT: 1 | 2 | 5 | 9
53. ANS:
a.
The shaded regions in the area model above correspond to exactly one of the two people belonging to a gym.
So, P(only one person belongs to gym) =
b. P(all three belong to gym) =
.
=
c. Austin’s calculations are most likely not correct because it is unlikely that gender and belonging to a gym
are independent; males are more likely to belong to a gym than are females.
PTS: 1
REF: Lesson 8-1
OBJ: 8-1.1 Use an area model to find the probability that two independent events both occur. | 8-1.3 Find
conditional probabilities and determine if two events are independent.
NAT: 1 | 2 | 5 | 9
54. ANS:
a.
b.
c.
(female and under 65) =
d. P(female or under 65) =
e. P(female | under 65) =
f. P(under 65 | female) =
g. They are not independent.
P(under 65) P(under 65 | female)
PTS: 1
REF: Lesson 8-1
OBJ: 8-1.3 Find conditional probabilities and determine if two events are independent. | 8-1.4 Use the
Multiplication Rule to find the probability that two events both occur when the events are not independent.
NAT: 1 | 2 | 5 | 9
55. ANS:
a. (0.12)(200) = 24 people
b. (0.12)(70) = 8.4 people
PTS: 1
REF: Lesson 8-4
OBJ: 8-4.1 Review and synthesize the major objectives of the unit.
NAT: 1 | 2 | 5
56. ANS:
a.
b. They should expect to make money. The expected cost of each spin over the long run is 5(0.125) + 10(0.5)
+ 25(0.25) + 50(0.125) = 18.125. Since they are charging $20, they will make an average of 20 – 18.125 =
$1.875 per spin. Students might assume a certain number of spins and compare money collected to value of
prizes awarded. They will get the same answer.
c. The fair price is $18.125.
d. Since they will spend an average of $18.125 per spin, they will need to collect a total of
500(18.125) + 2,000 = $11,062.50. Since 500 will pay to spin, they should charge
$22.13
PTS: 1
REF: Lesson 8-2
OBJ: 8-2.1 Compute the fair price (expected value) of insurance and games of chance. | 8-2.3 Compute the
expected value of a probability distribution using the formula.
NAT: 1 | 2 | 5
57. ANS:
a.
b.
PTS: 1
REF: Lesson 8-2
OBJ: 8-2.1 Compute the fair price (expected value) of insurance and games of chance.
NAT: 1 | 2 | 5