GUIDED PRACTICE
Vocabulary Check
✓
Concept Check
✓
3 APPLY
1
?㛭㛭㛭
1. The volume of a cone with radius r and height h is ᎏᎏ the volume of a 㛭㛭㛭㛭㛭
3
with radius r and height h. cylinder
Do the two solids have the same volume? Explain your answer. 2, 3. See margin.
2.
3.
3x
h
r
Skill Check
✓
4.a. 25 cm2
x
y
5.
ADVANCED
Day 1: pp. 755–756 Exs. 8–22
Day 2: pp. 756–758 Exs. 23–51,
Quiz 2 Exs. 1–7
6.
4 ft
4 cm
14 m
2
16π
3
121兹3苶
6.a. } ≈ 52.4 m2
4
5 cm
b. }} ≈ 16.8 ft 3
847兹3苶
6
b. } ≈ 244.5 m3
BLOCK SCHEDULE
11 m
2 ft
5 cm
7. CRITICAL THINKING You are given the radius and the slant height of a right
cone. Explain how you can find the height of the cone. The height of the cone is
the length of one leg of a right ¤ with hypotenuse equal to the slant height and one
leg the radius of the base, so use the Pythagorean Theorem to determine the height.
PRACTICE AND APPLICATIONS
STUDENT HELP
Extra Practice
to help you master
skills is on p. 826.
FINDING BASE AREAS Find the area of the base of the solid.
8.
9.
12.2 ft
10. Regular
base
10.1 in.
9 in.
3721π
}} ≈ 116.9 ft 2
100
90.9 in.2
18 mm
486兹3苶 ≈ 841.8 mm2
VOLUME OF A PYRAMID Find the volume of the pyramid. Each pyramid
has a regular polygon for a base.
67,183兹3苶
245
}} ≈ 155.2 ft 3
3
}} ≈ 81.7 m3
750
400
cm
3
11.
12.
13.
5m
12 cm
HOMEWORK HELP
Exs. 11–16
Exs. 17–19
Exs. 20–22
Exs. 23–28
Ex. 29
14.
9.2 ft
15.
16.
14.2 mm
18 in.
pp. 755–756 Exs. 8–22 (with 12.4)
pp. 756–758 Exs. 23–37, 40–51,
Quiz 2 Exs. 1–7 (with 12.6)
EXERCISE LEVELS
Level A: Easier
8–10, 40–47
Level B: More Difficult
11–33, 35–37, 48–51
Level C: Most Difficult
34, 38, 39
HOMEWORK CHECK
To quickly check student understanding of key concepts, go over
the following exercises: Exs. 10,
14, 18, 20, 24, 25, 27, 32. See also
the Daily Homework Quiz:
• Blackline Master (Chapter 12
Resource Book, p. 80)
•
Transparency (p. 91)
2. Yes; both have radius r and height
1
h, so both have volume }}␲r 2h.
3
3. Yes; the volume of the prism is
y ⴢ y ⴢ x = xy 2; the volume of the
7m
STUDENT HELP
Example 1:
Example 2:
Example 3:
Example 4:
Example 5:
12.7 ft
7m
10 cm
Day 1: pp. 755–756 Exs. 8–22
Day 2: pp. 756–758 Exs. 23–37,
40–51, Quiz 2 Exs. 1–7
Day 1: pp. 755–756 Exs. 8–22
Day 2: pp. 756–758 Exs. 23–37,
40–51, Quiz 2 Exs. 1–7
y
In Exercises 4–6, find (a) the area of the base of the solid and (b) the
volume of the solid.
4.
BASIC
AVERAGE
y
r
y
100
b. }} ≈ 33.3 cm3
3
5.a. 4π ≈ 12.6 ft
ASSIGNMENT GUIDE
1
3
pyramid is }} ⴢ y 2 ⴢ 3x = xy 2.
20 cm
14 in.
294兹3
苶 ≈ 509.2 in.3
10 mm
710兹3苶 ≈ 1229.8 mm3
12 cm
1440兹3苶 ≈ 2494.2 cm3
12.5 Volume of Pyramids and Cones
755
755
VOLUME OF A CONE Find the volume of the cone. Round your result to
two decimal places.
526.27 cm3
667.06 in.3
3
17. 48.97 ft
18.
19.
11.5 cm
COMMON ERROR
EXERCISE 17 Some students
may multiply the base area by the
slant height instead of by the height.
Remind them that the slant height is
used in the formula for the surface
area of a cone, but that the height
is used in the volume formula.
Encourage students to write out
formulas to avoid mistakes.
6 ft
13 in.
15.2 cm
7 in.
3 ft
xy USING ALGEBRA Solve for the variable using the given information.
20. Volume = 270 m3
10 m
STUDENT HELP NOTES
Homework Help Students
can find help for Exs. 23–25 at
www.mcdougallittell.com. The
information can be printed out
for students who don’t have
access to the Internet.
21. Volume = 100π in.3
5 in.
22. Volume = 5兹3
苶 cm3
5 cm
h
9m
r
INT
STUDENT HELP
NE
ER T
HOMEWORK HELP
Visit our Web site
www.mcdougallittell.com
for help with Exs. 23–25.
2兹3 cm
12 in.
COMPOSITE SOLIDS Find the volume of the solid. The prisms, pyramids,
and cones are right. Round the result to two decimal places.
23.
24.
25.
6 ft
2.3 cm
5.1 m
6 ft
2.3 cm
6 ft
3.3 cm
6 ft
5.1 m
5.1 m
16.70 cm3
97.92 m3
288 ft 3
AUTOMATIC FEEDER In Exercises 26 and 27, use the diagram of the
automatic pet feeder. (1 cup = 14.4 in.3)
26. Calculate the amount of food that can be
placed in the feeder. about 12 c
27. If a cat eats half of a cup of food, twice per day,
will the feeder hold enough food for three days? yes
28.
29.
756
756
ANCIENT CONSTRUCTION Early
civilizations in the Andes Mountains in Peru
used cone-shaped adobe bricks to build homes.
Find the volume of an adobe brick with a
diameter of 8.3 centimeters and a slant height of
10.1 centimeters. Then calculate the amount of
space 27 of these bricks would occupy in a mud
mortar wall. about 166 cm3; about 4482 cm3
2.5 in.
7.5 in.
4 in.
SCIENCE CONNECTION During a chemistry lab, you use a funnel to pour a
solvent into a flask. The radius of the funnel is 5 centimeters and its height
is 10 centimeters. If the solvent is being poured into the funnel at a rate of
80 milliliters per second and the solvent flows out of the funnel at a rate of
65 milliliters per second, how long will it be before the funnel overflows?
(1 mL = 1 cm3) about 17.5 sec
Chapter 12 Surface Area and Volume
FOCUS ON
CAREERS
USING NETS In Exercises 30–32, use the net to sketch the solid. Then find
the volume of the solid. Round the result to two decimal places.
116.08 m3
30.
31.
32.
4m
CAREER NOTE
EXERCISE 33
Additional information about
volcanologists is available at
www.mcdougallittell.com.
10 cm
6 cm
5 ft
16 m
301.59 cm3
STUDENT HELP NOTES
14.73 ft 3
33. FINDING VOLUME In the diagram at the right,
RE
FE
L
AL I
VOLCANOLOGY
Volcanologists
collect and interpret data
about volcanoes to help
them predict when a
volcano will erupt.
INT
a regular square pyramid with a base edge of
4 meters is inscribed in a cone with a height of
6 meters. Use the dimensions of the pyramid to
find the volume of the cone. 16π ≈ 50.3 m3
34.
NE
ER T
CAREER LINK
www.mcdougallittell.com
36. No; it takes the sand
slightly more than an
hour (about 62 min) to
fall from one part of the
glass to the other.
Look Back As students look back
to p. 466 for help with finding geometric means, remind them that
the geometric mean of two numbers derives from a proportion:
the first number is to the geometric mean as the geometric mean is
to the second number.
6m
4m
r
VOLCANOES Before 1980, Mount St. Helens
was cone shaped with a height of about 1.83 miles
and a base radius of about 3 miles. In 1980,
Mount St. Helens erupted. The tip of the cone was
destroyed, as shown, reducing the volume by
0.043 cubic mile. The cone-shaped tip that was
destroyed had a radius of about 0.4 mile. How
tall is the volcano today? (Hint: Find the height of
the destroyed cone-shaped tip.) about 1.58 mi
30.
5 ft
31.
10 cm
Test
Preparation
37. Assume that half the
sand has accumulated
after 30 min. Let r be
the radius of the pile
containing all the sand
and r1 the radius of the
pile containing half.
冉 冊
1
1 1
}}π(r1)3 = }} }}πr 3 ,
3
2 3
1
3
so (r1) = }}(r 3) and
2
1
r1 = 3 }} • r.
2
冪莦
★ Challenge
MULTI-STEP PROBLEM Use the diagram of the hourglass below.
6 cm
35. Find the volume of the cone-shaped pile of sand.
about 62.12 in.3
36. The sand falls through the opening at a rate of
32.
one cubic inch per minute. Is the hourglass a
true “hour”-glass? Explain. (1 hr = 60 min) See margin.
37.
8m
Writing
The sand in the hourglass falls into a
conical shape with a one-to-one ratio between
the radius and the height. Without doing the
calculations, explain how to find the radius and
height of the pile of sand that has accumulated
after 30 minutes.
4m
3.9 in.
3.9 in.
FRUSTUMS A frustum of a cone is the part of the cone that lies between
the base and a plane parallel to the base, as shown. Use the information
below to complete Exercises 38 and 39.
One method for calculating the volume of a frustum
is to add the areas of the two bases to their geometric
1
3
mean, then multiply the result by ᎏᎏ the height.
STUDENT HELP
Look Back
For help with finding
geometric means, see
p. 466.
ADDITIONAL PRACTICE
AND RETEACHING
2 ft
38. Use the measurements in the diagram to calculate
the volume of the frustum. 156π ft 3
For Lesson 12.5:
9 ft
6 ft
39. Write a formula for the volume of a frustum that
has bases with radii r1 and r2 and a height h.
1
3
V = }} (r12 + r22 + r1r2)hπ
12.5 Volume of Pyramids and Cones
757
• Practice Levels A, B, and C
(Chapter 12 Resource Book,
p. 69)
• Reteaching with Practice
(Chapter 12 Resource Book,
p. 72)
•
See Lesson 12.5 of the
Personal Student Tutor
For more Mixed Review:
•
Search the Test and Practice
Generator for key words or
specific lessons.
757
4 ASSESS
MIXED REVIEW
FINDING ANGLE MEASURES Find the measure of each interior and exterior
angle of a regular polygon with the given number of sides. (Review 11.1)
DAILY HOMEWORK QUIZ
40. 9 140°, 40°
Transparency Available
1. Find the volume of the pyramid
with the regular base.
7
11
4
11
43. 22 163 }}°, 16 }}°
1
19
18
19
41. 10 144°, 36°
42. 19 161}}°, 18}}°
44. 25 165.6°, 14.4°
45. 30 168°, 12°
FINDING THE AREA OF A CIRCLE Find the area of the described circle.
(Review 11.5 for 12.6)
625π
4
46. The diameter of the circle is 25 inches. }} ≈ 490.87 in.2
15 in.
26,569π
100
47. The radius of the circle is 16.3 centimeters. }} ≈ 834.69 cm2
48. The circumference of the circle is 48π feet. 576π ≈ 1809.56 ft 2
8 in.
49. The length of a 36° arc of the circle is 2π meters. 100π ≈ 314.16 m2
80兹3苶 ≈ 138.56 in.
3
USING EULER’S THEOREM Calculate the number of vertices of the solid
using the given information. (Review 12.1)
2. A right cone has a radius of
6 ft and a slant height of 9 ft.
What is its volume? Round the
result to two decimal places.
50. 32 faces; 12 octagons and 20 triangles
48 vertices
51. 14 faces; 6 squares and 8 hexagons
24 vertices
252.89 ft 3
3. A cone has a diameter of 10 m
and a volume of 150␲ m 3.
What is its height? 18 m
4. Find the volume of the solid.
192 cm 3
QUIZ 2
10 cm
Self-Test for Lessons 12.4 and 12.5
In Exercises 1–6, find the volume of the solid. (Lessons 12.4 and 12.5)
350π ≈ 1099.56 cm3
1. 1080 in.3
2. 1020 ft 3
3.
10 cm
6 cm
6 cm
15 ft
6 in.
18 in.
EXTRA CHALLENGE NOTE
Challenge problems for
Lesson 12.5 are available in
blackline format in the Chapter 12
Resource Book, p. 76 and at
www.mcdougallittell.com.
10 in.
243π
}} ≈ 190.85 m3
4. 4
5.
6.
36 mm
147兹3苶
} ≈ 63.65 in.3
4
9 in.
4.5 m
42 mm
3
7 in.
21,168 mm
7.
758
758
17 ft
8 ft
9m
ADDITIONAL TEST
PREPARATION
1. WRITING Cone B has the same
height as Cone A, but twice the
base diameter. Cone C has the
same base diameter as Cone A,
but twice the height. Do Cones B
and C have the same volume?
Explain. See right.
ADDITIONAL RESOURCES
An alternative Quiz for Lessons
12.4 and 12.5 is available in the
Chapter 12 Resource Book, p. 77.
14 cm
STORAGE BUILDING A road-salt
storage building is composed of a regular
octagonal pyramid and a regular octagonal
prism as shown. Find the volume of salt that
the building can hold. (Lesson 12.5) about 5633 ft 3
11 ft
8 ft
10 ft
Chapter 12 Surface Area and Volume
Additional Test Preparation Sample answer:
1
1. No; Sample answer: If the volume of Cone A is }}␲r 2h,
1
1
3
then the volume of Cone B is }}␲ (2r) 2h = 4 ⴢ }}␲ r 2h,
3
3
because the radius is twice that of A. The volume of
1
1
Cone C is }}␲ r 2 (2h ) = 2 ⴢ }}␲ r 2h, because the height
3
3
is twice that of A. So, Cone B has twice the volume of
Cone C.