Unit 4 Assessment • Circles and Volume 1.
__
In circle O, which term does not describe PR ?
3.
Q
A cross section of this triangular prism is taken perpendicular to the prism’s triangular base.
R S O P T A. chord
What is the shape of the cross section?
B. diameter
A. rectangle
C. radius
B. triangle
D. secant
C. trapezoid
2.
Identify the solid that will be formed by rotating the figure 360° about the dashed line. 4.
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Unit 4
D. ellipse
Jane drew several circles that have different radii, using a compass. What must be true of the circles she drew? A. All of the circles are similar but not necessarily congruent.
A. triangular prism B. sphere C. cylinder D. cylinder with two cones removed
B. All of the circles are congruent but not necessarily similar. C. All of the circles are similar and congruent to one another. D. None of the circles are similar or congruent to one another.
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5.
The sphere shown has a radius of 12 meters. What is the volume of the sphere?
12 m
A. 288p cubic meters B. 576p cubic meters C. 1,296p cubic meters D. 2,304p cubic meters
6.
Two cylinders are shown below. Each has a height of 10 units and a radius of r units, and each has been cut by a horizontal plane that is parallel to the bases of both cylinders.
r
r
10 Right Cylinder
Oblique Cylinder
A. The right cylinder has a greater volume than the oblique cylinder because its cross section has a greater area. B. The oblique cylinder has a greater volume than the right cylinder because its cross section has a greater area. C. Both cylinders have the same volume because their cross sections have the same area. D. It is not possible to determine which cylinder has a greater volume because the length of the radius, r, is not given.
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Which statement accurately compares the volumes of these two cylinders?
7. Figure A shows a square inscribed in a circle. Figure B shows a decagon inscribed in a circle.
9.
A cross section is taken from a sphere. Which of the following is true? A. The cross section must be a circle. B. The cross section must be an ellipse. C. The cross section may be a circle or an ellipse. D. The cross section may be a circle or a line.
Figure A
Figure B p
Aviva wants to use a polygon inscribed in a circle to approximate the area of the circle. Which polygon described below has an area that most closely approximates the area of the circle in which it is inscribed?
10. In circle O, u 5 __ 3 and OM 5 9 inches.
N
M
A. a square B. a decagon
9 in.
O
C. a triangle Unit 4
D. a regular polygon with 12 sides
What is the length of MN ? 8.
The Pyramid of Cestius is an ancient pyramid in Rome, Italy. It has a height of 37 meters and a square base with sides measuring 30 meters.
p A. __ 27 inches
B. 3p inches C. 6p inches
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D. 18p inches
37 m
30 m What is the volume of the pyramid? A. 5,550 cubic meters B. 11,100 cubic meters C. 13,690 cubic meters D. 33,300 cubic meters
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11.
Pablo wants to draw two tangent lines to circle A from point B. He started by drawing a line from A to B. He then constructed line CD.
13. The napkin ring has a height of 3 centimeters and a thickness of 0.2 centimeter. The diameter of the hole in the center is 3.8 centimeters.
3.8 cm
C
0.2 cm
B E
A
3 cm
D
What should Pablo do next? ___
___
A. Construct BC and BD . B.
___ ___ . Construct AC and AD
___
C. Construct circle B with radius BE . ___
D. Construct circle E with radius BE .
If the napkin ring is made of solid silver, approximately what is the volume of silver in the napkin ring? A. 0.4 cubic centimeters B. 7.5 cubic centimeters C. 37.7 cubic centimeters
12. The circle shows intersecting chords.
D. 113.0 cubic centimeters
D
2x
C
4
12 8 B
What length is represented by x? A. 3 units B. 4 units C. 6 units D. 8 units
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A
14. Circle G is shown.
16. The cup dispenser at a water cooler contains cone-shaped paper cups with the dimensions shown.
6 cm
F 30°
G
H 8 cm
J
What is the measure of FH ? Approximately how many cubic centimeters of water can each cup hold?
A. 15° B. 30°
A. 75 cubic centimeters
C. 60°
B. 201 cubic centimeters
D. 330°
C. 226 cubic centimeters D. 301 cubic centimeters Unit 4
15. Line segments AC and BC are radii of circle C. Points A and B are connected to form two vertices of n ABC. The radius of circle C is 12 cm long, and m/ ACB 5 60°.
A B
12 cm Duplicating this page is prohibited by law. © 2016 Triumph Learning, LLC
60° C
___
The shaded area bounded by AB and AB is called a segment of the circle. Which best approximates the area of this segment? A. 6 square centimeters B. 13 square centimeters C. 62 square centimeters D. 75 square centimeters
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17.
___
In circle O, /QMP measures 41°.
M
__
are intersecting chords. 19. In the circle, FG and HJ
J
N
36° G
O
z
F
Q
48°
P
H
Which of the following must also be true? A. m/QNP 5 20.5°; m/QOP 5 20.5° B. m/QNP 5 41°; m/QOP 5 20.5° C. m/QNP 5 41°; m/QOP 5 82° D. m/QNP 5 82°; m/QOP 5 82°
18. Jonah accurately constructed point Z for nWXY, as shown.
W
What is the value of z? A. 6° B. 24° C. 42° D. 44°
20. The diameter of a circle is 8 meters. If the line of the diameter is extended 2 meters beyond the circle to point P and a tangent drawn from point P ___ intersects the circle at point B, about how long is PB ? A. 4.0 meters
Z
B. 4.5 meters
Y
Which of the following is true of this construction?
C. 8.0 meters D. 8.9 meters
› __› ___ YZ . is the perpendicular bisector of WX __
A. YZ is the angle bisector of /X. B.
C. Point Z is the center of the circle that can be circumscribed about nWXY. D. Point Z is the center of the circle that can be inscribed in nWXY.
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X
21. Point P is outside circle Q. Construct two lines through point P that are tangent to circle Q. Show all work.
Q
P
22. A silo used to hold grain consists of a cylinder with a height of 25 meters capped by a hemisphere with a diameter of 6 meters.
Unit 4
6m
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25 m
What is the approximate volume of the silo to the nearest cubic meter? Show all work.
V<
cubic meters
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23. At Camp Coyote, campers from Cabins A, B, and C will compete in a scavenger hunt. To make the game fair, the counselors tell the campers that the first clue will be placed at a location that is equidistant from all three cabins. The three cabins form the vertices of a triangle, as shown on the map.
A
B
C
A.
The leader of Cabin A, Alicia, says the circumcenter of nABC will show the location of the first clue. The leader of Cabin B, Bella, says the incenter will show the location. Who is correct—Alice, Belinda, neither, or both? Explain.
B.
Construct and label a point on the diagram to indicate the correct location of the first clue.
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C. Show that the point you drew is correct by drawing a circle with that point as the center. Explain how the circle you drew shows that your point is correct.
24. Regular octagon ABCDEFGH is drawn around circle O.
B
C
A
D
O H
G A.
E
r
s
F
A polygon with n sides is sometimes called an n-gon. What is an expression for the perimeter (P) of an n-gon if each side has a length of s units?
P5
Write an expression for the area of FGO in terms of s and r.
B.
Unit 4
5 A tri An n-gon can be broken up into n triangles like FGO. What is an expression for the total area of an n-gon in terms of n, s, and r?
A 5 n-gon
What is an expression for the area of an n-gon in terms of its perimeter (P) and r?
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5 A n-gon C. For very large values of n, the perimeter of an n-gon would be about equal to the circumference of the circle around which it is drawn. Substitute 2pr for P in your equation to estimate the area of the n-gon. An-gon 5 D.
The formula you wrote in part C also can be used to find the area of another kind of geometric figure. Name the figure.
Go On 45
25. The right triangle below will be rotated 360° about line a to form solid A. Then, the same right triangle will be rotated 360° about line b to form solid B.
a
b
5 in. 3 in.
A.
What type of solid is solid A? What type of solid is solid B? Predict which solid will have the greater volume.
B.
Find the volumes of both solids. Show all work and give your answers in terms of p.
C. How much greater is the volume of the larger solid? Was your prediction correct?
STOP 46
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4 in.
Unit 5 Assessment • Geometric and Algebraic Connections 1.
Which shows the correct center and radius for the circle with this equation?
(
1
)
2
9
(x 2 3)2 1 y 1 __ 2 5 __ 16
C. D.
Which statement is true of the circle and the line described by the nonlinear system shown below? (x 1 1)2 1 (y 1 1)2 5 25
3 ( 1) 9 1 2 ); radius: __ 16 center: ( 3, 2__ 3 1 2 ); radius: __ 4 center: ( 23, __ 9 1 2 ); radius: __ 16 center: ( 23, __
2 ; radius: __ 4 A. center: 3, 2__ B.
3.
x 1 y 5 3 A. The line is tangent to the circle. B. The line is a secant of the circle. C. The line does not intersect the circle. D. The line passes through the center point of the circle.
___
2. Point B is the midpoint of CD on the grid below. y
4. Points P (22, 1) and Q (4, 2) are the endpoints of the line segment that is graphed below.
8
y
6 4 2
–6
–4
C
–2
B
2
4
6
8
x
6 4
–2
Q
2
–4
P
–6
–8
–6
–4
–2
0
2
4
6
8
x
Unit 5
–8
8
D
–2
–8
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–4 –6
Which expression should be used to determine the coordinates of point B?
( 3 1 (22)
2 1 (21)
)
A. _______ , _______ 2 2 B. (2 1 (23), 1 1 (22))
( 2 1 (23) 1 1 (22) ) 23 1 (22) ____ 211 ( ________ , 2 ) 2
C. _______ , _______ 2 2 D.
–8
Which expression should be used to determine ___ the length of PQ ? __________________
A. √ (4 2 (22))2 1 (2 2 1)2 __________________
(4 2 (22)) 2 (2 2 1) B. √ 2
2
__________________
(4 1 (22)) 1 (2 1 1) C. √ 2
2
__________________
(4 1 (22)) 2 (2 1 1) D. √ 2
2
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5. Circle A has a radius of 3 centimeters. Circle B has a radius of 6 centimeters. For each circle,
7.
Use the line that is graphed below to answer the question.
find the ratio of its circumference to its diameter
y
( __ dC ). Which statement is true?
8
A. The ratio of circle A’s circumference to its diameter is less than the ratio of circle B’s circumference to its diameter. B. The ratio of circle A’s circumference to its diameter is half the ratio of circle B’s circumference to its diameter.
6 4 2
–8
–6
–4
–2
2
4
6
8
x
–2 –4
C. The ratio of circle A’s circumference to its diameter is greater than the ratio of circle B’s circumference to its diameter. D. The ratio of circle A’s circumference to its diameter is the same as the ratio of circle B’s circumference to its diameter.
0
–6 –8
Which equation represents a line that is parallel to the line that is graphed above? 2
A. y 5 2 __ 3 x 2 4 3
6.
A wheelchair accessible ramp must have an angle of elevation of no more than 8°. About what is the length, c, of a ramp needed to reach a doorway that is 4 feet off the ground? About what is the distance, b, from the start of the ramp to the doorway?
B. y 5 __ 2 x 1 3 2
C. y 5 __ 3 x 2 1 3
D. y 5 2 __ 2 x 1 7
8. Point J, located at (3, 8), lies on a circle with center (6, 4). What is the equation of the circle? A. (x 2 3)2 1 (y 2 8)2 5 5 8° b
4 ft
A. c < 114.68 ft; b < 114.4 ft
2 2 B. (x 2 3) 1 (y 2 8) 5 25 2 2 C. (x 2 6) 1 (y 2 4) 5 5 2 2 D. (x 2 6) 1 (y 2 4) 5 25
B. c < 114.4 ft; b < 114.68 ft C. c < 28.74 ft; b < 28.46 ft D. c < 28.46 ft; b < 28.74 ft
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c
9.
A triangle, nXYZ, is graphed on the coordinate plane below.
11.
What is the distance between points E(25, 23) and F (2, 27)? __
A. √11
y
___
B. √29
10 9 8 7 6 5 4 3 2 1
___
C. √33 ___
D. √65
Y
X
0
Z
12. Use the line that is graphed below to answer the question.
x
1 2 3 4 5 6 7 8 9 10
Which of the following proves that the triangle is isosceles?
y 8
A. m/YZX 5 45°
6
B. XZ 5 6
4 2
C. m/XYZ . m/ZXY –8
D. XY 5 YZ
–6
–4
–2
0
2
4
6
8
x
–2 –4 –6
10. Points A(22, 9) and B(6, 1) are the endpoints of the line segment shown below.
–8
y
Which equation represents a line that is perpendicular to the line shown above?
10
4
A. y 5 __ 3 x 1 2
8 6
3
B. y 5 __ 4 x 2 8
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4 2
B –10
–8
–6
–4
–2
0
Unit 5
A
2
4
6
8
10
x
4
C. y 5 2 __ 3 x 1 7 3
D. y 5 2 __ 4 x 2 2 3
Which point is __ 4 of the distance from point A to point B? A. (0, 7) B. (4, 3) C. (4, 4) D. (2, 5)
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13. Use the line that is graphed below to answer the question.
15. Use the line that is graphed below to answer the question.
y
y
8
8
6
6
4
4
2
–6
–4
–2
0
2
4
6
8
–8
–6
–4
–2
–2
–2
–4
–4
–6
–6
–8
–8
Which equation represents a line that passes through the point (7, 4) and is parallel to the line that is graphed above?
A. y 5 2x 1 5
B. y 5 2 __ 7 x 1 5
1
B. y 5 2x 2 3
C. y 5 7x 1 4
C. y 5 2 __ 2 x 2 5
D. y 5 7x 2 45
D. y 5 2 __ 2 x 2 1
14. A cylindrical salt shaker has a radius of 2 cm and a height of 7 cm. It comes in a box that is 8 cm tall and has a square base with an edge length of 4 cm. Which is closest to the volume of air in the box outside the salt shaker?
B. 40 cm3 C. 61 cm3 D. 84 cm3
2
4
6
8
x
Which equation represents a line that passes through the point (24, 23) and is perpendicular to the line that is graphed above?
1
A. y 5 2 __ 7 x 1 3
A. 22 cm3
0
1 1
16. A biologist is evaluating a soil sample for earthworms. In 6 cubic inches of soil, the biologist finds 11 earthworms. If the density of earthworms in this soil sample is representative of the density of earthworms in a nearby field, which most accurately describes the density of earthworms in the field? A. 22 earthworms per cubic foot of soil B. 88 earthworms per cubic foot of soil C. 264 earthworms per cubic foot of soil D. 3,168 earthworms per cubic foot of soil
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–8
2
x
17.
Which shows the graph of the circle with this equation? (x 1 3)2 1 (y 1 1)2 5 4 y
A.
–6
–4
–2
6
6
4
4
2
2
0
2
4
6
x
–6
–4
–2
–6
–4
–2
0
–2
–2
–4
–4
–6
–6
y
B.
y
C.
6
4
4
2
2
2
4
6
x
4
6
x
2
4
6
x
y
D.
6
0
2
–6
–4
–2
–2
0 –2
–4
–4
–6
–6
Unit 5
18. A floodlight in a storage facility spreads light over a distance of 30 feet and is set to rotate through an angle of 150°.
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30 ft 150°
About how many square feet of the facility will be lit by the floodlight? A. 79 square feet B. 236 square feet C. 1,178 square feet D. 5,887 square feet
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19. The triangle that is graphed in the coordinate plane is isosceles.
A. neither
y 8
B. parallel
M
C. perpendicular
6
L
4
–6
D. cannot be determined
2
X –8
1
20. Are the lines y 5 24x 1 1 and y 5 2 __ 4 x 2 3 parallel, perpendicular, or neither?
–4
N
–2
0
2
4
6
8
x
–2 –4 –6 –8
___
Which sentence proves that MX bisects the base of the triangle? ___
A. MX bisects the base because /NXM is a right angle. ___
B. MX bisects the base because m/MNL 5 m/NLM. ___
C. MX bisects the base because LX 5 XN 5 4 units. ___
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base because D. MX bisects the __ LX 5 XN 5 √ 8 units.
21. The equation of a circle in general form is given below. x2 1 y2 2 4x 1 6y 1 9 5 0 Convert this equation to standard form by completing the square. Identify the circle’s center and radius. Show all work.
standard form: center: (
,
)
radius:
22. Two lines are graphed on the coordinate plane below. y 8 6
(0, 3)
Unit 5
4
(–4, 3)
2
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–8
–6
–4
–2
0
2
4
6
8
x
–2
(–1, –3)
–4 –6
(5, –7)
–8
Find the slope of each line, and explain why the lines are or are not parallel.
Go On 53
23. In circle O, /MNP is an inscribed angle that intercepts MQP . Use Circle O to prove that a 90° inscribed angle intercepts a semicircle. y 8
Q
6 4
M –4
–2
0
P
O
2
2
4
6
8
x
N –2 –4
____
___
A.
Calculate the slopes of MN and NP . Use those slopes to prove that the inscribed angle, /MNP, measures 90°. Explain and show all work.
slope of MN 5
slope of NP 5
B.
Explain why MP is a diameter of circle O.
____ ___
is a semicircle. C. Explain how you know that MQP
What does this indicate about the arc that a 90° inscribed angle intercepts?
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___
24. Two lines in a plane can be parallel, perpendicular, or intersecting but non-perpendicular. A. Line a passes through the points (2, 22) and (3, 2). Line b passes through the points (24, 6) and (0, 5). Determine if the lines are perpendicular. Show all work.
B.
Graph the two lines to check your work. y 10 8 6 4 2
–10
–8
–6
–4
–2
0
2
4
6
8
10
x
–2 –4 –6 –8 –10
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Unit 5
C. Justify your answer to part A, using the concept of rate of change.
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25. A dock that is anchored in the middle of a lake is 20 feet wide and 30 feet long. The bow (front end) of a kayak is tied to a corner of the dock by a rope that is 12 feet long, as shown.
20 ft
30 ft
Dock
bow 12 ft
Use your compass to draw the section of the surface of the lake over which the bow of the kayak may drift. Use letters to name angles and endpoints that are part of your drawing. Describe the geometric figure you drew, and explain what it indicates about where the bow of the kayak may drift.
Go On 56
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A.
B.
What is the approximate area, in square feet, of the surface of the lake over which the bow of the kayak may drift? Show all work.
A <
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Unit 5
C. If the 12-foot rope is replaced with a rope that is 10 feet long, by about how many square feet will the area of the surface of the lake over which the bow of the kayak may drift be decreased? Show and explain all work.
STOP 57
Unit 6 Assessment • Applications of Probability 1.
Consider the universal set, U, and its subsets, A and B.
3.
Eight tiles are in a fishbowl as shown.
U 5 {1, 2, 4, 6, 7, 10} A 5 {2, 4, 6, 10} B 5 {2, 6, 7}
S
A
V
A
N
N
A
H
Which set represents A > B? Jack will reach into the bag and select a tile without looking. Which represents the probability that he will select a tile that displays the letter N?
A. {2} B. {2, 6} C. {7, 10} D. {2, 4, 6, 7, 10}
1 B. __ 4
A sample of 100 full-time students at a local community college participated in a survey investigating whether students pay for their own tuition and whether students are employed to work while taking classes. The results of the survey are recorded in the table below. Pays Own Tuition
Does Not Pay Own Tuition
Employed
28
32
Not Employed
6
34
Which of the following conclusions can be drawn from the results of survey? A. Most students who are employed pay for their own tuition. B. Most students who pay for their own tuition are employed.
3 C. __ 8 1 D. __ 2
4.
If events A and B are independent, which statement about events A and B must be true? A. The outcome of event A does not affect the outcome of event B, so P(A > B) 5 P(A)P(B). B. The outcome of event A affects the outcome of event B, so P(A > B) 5 P(A)P(B). C. The outcome of event A does not affect the outcome of event B, so P(A < B) 5 P(A) 1 P(B). D. The outcome of event A affects the outcome of event B, so P(A < B) 5 P(A) 1 P(B).
C. Most students who pay for their own tuition and are employed have better salaries than those who do not pay for their own tuition and are employed in similar occupations. D. Most students who do not pay for their own tuition are not employed.
58
Go On
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2.
1 A. __ 8
5.
For two overlapping events A and B, which of the following is equal to P(A < B)? A. P(A) 1 P(B) B. P(A) 1 P(B) 2 P(A > B) C. P(A)P(B) P(A > B) D. ______ P(B)
6.
A large bag contains marbles of the following colors: black (B), green (G), red (R), and yellow (Y). A small bag contains marbles of the following colors: green (G), white (W), and yellow (Y). Which of the following represents the union of the sets of colors in the two bags?
8.
The conductor of a chorus records the names of three male soloists and three female soloists. The conductor writes each name on an index card. The index cards marked with male soloists’ names are placed in one hat, and the index cards marked with female soloists’ names are placed in another hat. The conductor will randomly choose one index card from each hat to select the pair that will perform a duet at the spring concert. From a given hat, each index card is equally likely to be chosen. The following tree diagram shows all the possible combinations for the duet.
Male Soloist Caleb (C)
Female Soloist Delilah (D) Hazel (H)
A. {G, R}
Simone (S)
B. {G, Y}
Delilah (D) Memphis (M)
C. {B, G, R, Y} D. {B, G, R, W, Y}
Hazel (H) Simone (S) Delilah (D)
The following Venn diagram illustrates __ __ P(C > D ) and P(C > D). Neither P(C > D) nor ______ P(C < D ) are displayed.
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U
C 0.4
D 0.1
?
? If events C and D are independent events, which represents P(D)? A. 0.03 B. 0.2
Teddy (T)
Hazel (H) Simone (S)
Which represents the probability that Memphis and Simone will be selected to perform the duet? 1 A. __ 3 1 B. __ 6
Unit 6
7.
1 C. __ 9 1 D. __ 12
C. 0.3 D. 0.5
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9.
The following Venn diagram describes the foreign languages being studied by members of the boys’ basketball team at Central High School. Students may choose to study French (F) or Spanish (S). Let the rectangle represent {U}, the universal set.
U
F Adam Bob Emesto Leon
Hayden
S Charlie Devon Fitz Gus Juan Keitaro Matt
11.
The universal set shown by the Venn diagram is the set of all possible outcomes of tossing a number cube with faces numbered 1 to 6. Subset O is the set of outcomes that are odd. Subset M is the set of outcomes that are multiples of 3.
U
2 4
O 1 3 5
M 6
______
Which contains all elements of { O > M } ? A. {3}
Which describes the elements of the subset __ {F > S }?
B. {2, 4}
A. players who study French but not Spanish
D. {1, 2, 4, 5, 6}
C. {1, 5, 6}
B. players who study Spanish but not French
D. players who study neither French nor Spanish
10. If event A is independent of event B, which of the following must be true? A. P(A | B) 5 P(A) 1 P(B) P(A)P(B)
B. P(A | B) 5 ______ P(B) C. P(A | B) 5 P(A < B) 2 P(A > B) P(A)
D. P(A | B) 5 ___ P(B)
12. The physical education teachers surveyed ninth-grade students to find out which field trip they were most interested in taking. Golf (G)
Swimming (S)
Boys (B)
72
24
Girls (G)
46
50
If a ninth-grade boy from the school is randomly chosen, which of the following would you expect to be true? A. He would be about 3 times as likely to prefer a golf trip to a swimming trip. B. He would be about 3 times as likely to prefer a swimming trip to a golf trip. C. He would be about 7 times as likely to prefer a golf trip to a swimming trip. D. He would be just as likely to prefer a golf trip as a swimming trip.
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C. players who study either French or Spanish but not both
13. Tenth-grade students will have to sign up for a health elective. A survey of a group of tenthgrade students asked which elective they would most like to take. First Aid (F)
Nutrition (N)
Boys (B)
20
5
Girls (G)
15
10
Which is equivalent to P(B < N)? A. 10% B. 35% C. 70% D. 85%
14. A white number cube, a red number cube, and a green number cube, each with faces numbered 1 to 6, are tossed at the same time. Which of the following represents the probability of all three cubes landing on odd numbers given that the white cube landed on 3? 1 A. __ 2
15. Students were surveyed to investigate whether they had achieved a score of 85 or higher on the U.S. history midterm and whether or not they had played computer games the night before the U.S. history midterm exam. The results of the survey are recorded in the following joint frequency table. Score $ 85
Score , 85
Played Computer Games
1
7
Did Not Play Computer Games
10
6
Which represents the probability that a randomly selected student from the class had a score greater than or equal to 85 given that the student did not play computer games the night before the exam? 5 A. __ 12 5 B. __ 8 2 C. __ 3 10 D. __ 11
1
B. __ 4
1
D. ___ 36
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1 C. __ 8
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16. The following Venn diagram shows the intersection of two sets. Set A shows the number of ninth-grade students in the school. Set B shows the number of students in the school band.
U
A
B
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12 16
8 Based on the Venn diagram, what is the value of P(B | A)? What does this probability represent? Show all of your work.
17.
Eleventh- and twelfth-grade students at Fairlawn High School were asked whether or not they have their own cars. The results of the survey are recorded in the following two-way frequency table. Have Car (C)
No Car (N)
Eleventh Grade (E)
8
32
Twelfth Grade (T)
22
18
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Suppose a student is randomly chosen from a list of eleventh- and twelfth-grade students at the high school. What is the probability that the student chosen will be in twelfth grade or have a car? Use probability and set notation to show or explain how you determined the answer.
18. Abby, Brian, and Chelle are spinning a spinner with 9 equal-size sections numbered 1 to 9. • If the spinner lands on a prime number, Abby gets a point. • If the spinner lands on an even number, Brian gets a point. • If the spinner lands on a number greater than 5, Chelle gets a point. A.
List the elements in each set.
The universal set:
U 5
The winning outcomes for Abby:
A 5
The winning outcomes for Brian:
B 5
The winning outcomes for Chelle:
B.
Use the triple Venn diagram provided below to classify each element of U in its appropriate subset. Use labels to highlight the universal set and sets A, B, and C.
Shade the subset of U that represents A < B < C. What does this region represent? Which elements of U are included in this subset?
_________
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C 5
C. Are there any empty regions in the diagram? Explain the meaning of empty subsets of U in the context of this game.
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19. Eighty tenth-grade students at Milton High School were asked which school lunch they like best. Of the boys, 25 chose pizza, 4 chose chicken, and 11 chose tacos. Of the girls, 7 chose pizza, 20 chose chicken, and 13 chose tacos. A.
Create a two-way frequency table for these data. Total Boys Girls Total
B.
What is the probability that a tenth-grade student chosen randomly at Milton High School likes pizza best? Explain how you determined your answer.
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C. What is the probability that a tenth-grade boy chosen randomly at Milton High School likes pizza best? Explain how you determined your answer.
20. Sixty cupcakes were prepared for a celebration. Each cupcake has blue, red, or white icing. The cake is either chocolate cake, yellow cake, or strawberry cake.
Icing
blue (B) 2 5
Combination
chocolate (C)
1 5
yellow (Y)
2 15
chocolate (C)
1 15 1 20
yellow (Y)
1 10
strawberry (S)
1 20
chocolate (C)
1 4
yellow (Y)
3 20
strawberry (S) 1 5
red (R)
2 5
white (W)
A.
Katie is the first to select a cupcake. Given that she has chosen a cupcake with blue icing, what is the probability that she chose each of the following?
a chocolate cupcake
a yellow cupcake
a strawberry cupcake
B.
Assuming Katie had chosen randomly (without choosing an icing color first), which two cupcake combinations had the same probability of being chosen? Explain your response.
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Cake
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