3 APPLY
ASSIGNMENT GUIDE
GUIDED PRACTICE
✓
Concept Check ✓
Vocabulary Check
BASIC
Day 1: pp. 762–765 Exs. 10–17,
20–29, 35, 36, 45–48, 50–57
AVERAGE
Day 1: pp. 762–765 Exs. 10–18,
20–29, 35–40, 45–48, 50–57
ADVANCED
Day 1: pp. 762–765 Exs. 10–18,
20–29, 35–40, 44–57
diameter of 10 millimeters. Explain her error(s).
2. She used the formula
for the area of a ›
with radius r rather
than the formula for
the volume of a sphere
with radius r, and used
the diameter rather
than the radius.
Skill Check
BLOCK SCHEDULE WITH 12.5
pp. 762–765 Exs. 10–18, 20–29,
35–40, 45–48, 50–57
EXERCISE LEVELS
Level A: Easier
10–12
Level B: More Difficult
13–43, 45–48, 50–57
Level C: Most Difficult
44, 49
HOMEWORK CHECK
To quickly check student understanding of key concepts, go
over the following exercises:
Exs. 12, 22, 24, 28, 36. See also
the Daily Homework Quiz:
• Blackline Master (Chapter 12
Resource Book, p. 96)
•
Transparency (p. 92)
?㛭㛭㛭 from a 㛭㛭㛭㛭㛭
?㛭㛭㛭 is called a sphere.
1. The locus of points in space that are 㛭㛭㛭㛭㛭
equidistant; point
2. ERROR ANALYSIS Melanie is asked to find the volume of a sphere with a
✓
V = πr 2
= π(10)2
10 mm
= 100π
In Exercises 3–8, use the diagram of the sphere, whose center is P.
3, 4. Sample answers are given.
Æ Æ
Æ
3. Name a chord of the sphere. QS , RT , or TS
Æ Æ
R
PQ , PR , or
4. Name a segment that is a radius of the sphere. Æ
PS
Æ q
5. Name a segment that is a diameter of the sphere. QS
6. Find the circumference of the great circle that
contains Q and S. 6π ≈ 18.85 units
7. Find the surface area of the sphere.
36π ≈ 113.10 sq. units
8. Find the volume of the sphere. 36π ≈ 113.10 cu. units
9.
P
T
S
CHEMISTRY A helium atom is approximately spherical with a radius of
about 0.5 ª 10º8 centimeter. What is the volume of a helium atom?
about 5.24 ª 10º25 cm3
PRACTICE AND APPLICATIONS
STUDENT HELP
Extra Practice
to help you master
skills is on p. 826.
FINDING SURFACE AREA Find the surface area of the sphere. Round your
result to two decimal places.
4071.50 cm2
186.27 ft 2
530.93 m2
10.
11.
12.
18 cm
6.5 m
7.7 ft
USING A GREAT CIRCLE In Exercises 13–16, use the sphere below. The
center of the sphere is C and its circumference is 7.4π inches.
13. What is half of the sphere called? a hemisphere
STUDENT HELP
HOMEWORK HELP
Example 1: Exs. 10–12
Example 2: Exs. 13–16
Example 3: Ex. 17
Example 4: Exs. 20–22,
41–43
762
762
14. Find the radius of the sphere. 3.7 in.
C
15. Find the diameter of the sphere. 7.4 in.
16. Find the surface area of half of the sphere.
27.38π ≈ 86.02 in.2
17.
SPORTS The diameter of a softball is 3.8 inches. Estimate the amount of
leather used to cover the softball. about 45.4 in.2
Chapter 12 Surface Area and Volume
FOCUS ON
APPLICATIONS
18.
PLANETS The circumference of Earth at the equator (great circle of
Earth) is 24,903 miles. The diameter of the moon is 2155 miles. Find the
surface area of Earth and of the moon to the nearest hundred. How does the
surface area of the moon compare to the surface area of Earth? See margin.
APPLICATION NOTE
EXERCISE 18 Additional information about planets is available at
www.mcdougallittell.com.
19. DATA COLLECTION Research to find the diameters of Neptune and its two
moons, Triton and Nereid. Use the diameters to find the surface area of each.
L
AL I
RE
INT
FE
PLANETS Jupiter
is the largest planet
in our solar system. It has a
diameter of 88,730 miles, or
142,800 kilometers.
See margin.
FINDING VOLUME Find the volume of the sphere. Round your result to two
decimal places.
3156.55 mm3
65.45 in.3
44,602.24 cm3
20.
21.
22.
2.5 in.
18.2 mm
22 cm
r 3 = 1000 m 3. When students take
the cube root of both sides, they
can treat m like a variable, so
3 3
兹苶
m 苶 = m, and r = 10 m. This
makes sense because the radius
is a one-dimensional length.
NE
ER T
APPLICATION LINK
www.mcdougallittell.com
USING A TABLE Copy and complete the table below. Leave your answers
in terms of π.
1372π
3
14π mm; 196π mm 2; }} mm3 23.
Radius of
sphere
Circumference
of great circle
Surface area
of sphere
Volume of
sphere
7 mm
?
㛭㛭㛭
?
㛭㛭㛭
?
㛭㛭㛭
6 in.; 12π in.; 288π in.3
24.
?
㛭㛭㛭
?
㛭㛭㛭
144π in.2
?
㛭㛭㛭
500π
5 cm; 100π cm ; }} cm 3
3
25.
?
㛭㛭㛭
10π cm
?
㛭㛭㛭
26.
?
㛭㛭㛭
?
㛭㛭㛭
?
㛭㛭㛭
?
㛭㛭㛭
4000π 3
ᎏᎏ m
3
2
10 m; 20π m; 400π m
2
b. 419.82 in.3
27.
28.
2
28. a. 2073.45 cm
29.
5.1 ft
18 cm
b. 7749.26 cm3
29. a. 375.29 ft
GRAPHING CALCULATOR
NOTE
EXERCISES 30–32 Students
can use the table features of a
graphing calculator to help with
Exercises 30 and 31 by setting the
table to an initial value of 1 with a
step of 1, and then entering the
formulas for volume and surface
area as the first two functions. The
third function can then be entered
from the variables list as the ratio
of the previously defined functions.
Students can graph the third
function directly for Exercise 32.
Students can verify that the graph
makes sense by writing the ratio
of the formulas explicitly and
simplifying.
COMPOSITE SOLIDS Find (a) the surface area of the solid and (b) the
volume of the solid. The cylinders and cones are right. Round your results
to two decimal places.
27. a. 488.58 in.2
9 in.
10 cm
12.2 ft
2
b. 610.12 ft 3
33. No; let r be the radius
of a sphere and S its
surface area. If the
radius is tripled, then
the surface area is
4π(3r)2 = 4π(9r 2) =
9(4πr 2) = 9S. The
surface area is
multiplied by 9.
(Another way to
consider this is to note
that surface area is a
function of r 2, not of r,
so tripling the value of r
multiplies the value of
S by 3 2 = 9.)
4.8 in.
TECHNOLOGY In Exercises 30º33, consider five spheres whose radii
are 1 meter, 2 meters, 3 meters, 4 meters, and 5 meters.
30. Find the volume and surface area of each sphere. Leave your results in
terms of π. See margin.
31. Use your answers to Exercise 30 to find the ratio of the volume to the
V
surface area, ᎏᎏ, for each sphere. }13}; }23}; 1; }43}; }53}
S
V
32. Use a graphing calculator to plot the graph of ᎏᎏ as a function of the
S
radius. What do you notice? Check graphs; the graph is linear.
33.
Writing
If the radius of a sphere triples, does its surface area triple?
Explain your reasoning. See margin.
12.6 Surface Area and Volume of Spheres
COMMON ERROR
EXERCISE 26 Dealing with the
units may be confusing for some
students when working backwards
from cubic meters. Simplifying the
4
4000␲
equation ᎏᎏ␲ r 3 = ᎏᎏ m 3 gives
3
3
763
18. about 197,402,900 mi 3, about
14,589,600 mi 3; the surface area
of the earth is about 13.5 times
the surface area of the moon.
(Answers may vary due to
rounding.)
19. The diameters of Neptune and
its moons Triton and Nereid are,
respectively, about 30,775 mi,
about 1680 mi, and about 211 mi.
Then the surface areas are
about 2,975,404,400 mi 2, about
8,866,800 mi 2, and 139,900 mi 2.
30. See next page.
763
34. VISUAL THINKING A sphere with radius r is inscribed in a cylinder with
height 2r. Make a sketch and find the volume of the cylinder in terms of r. 2πr 3
STUDENT HELP NOTES
30.
STUDENT HELP
INT
Homework Help Students can
find help for Exs. 35 and 36 at
www.mcdougallittell.com. The
information can be printed out for
students who don’t have access
to the Internet.
NE
ER T
HOMEWORK HELP
Visit our Web site
www.mcdougallittell.com
for help with Exs. 35
and 36.
xy USING ALGEBRA In Exercises 35 and 36, solve for the variable. Then
find the area of the intersection of the sphere and the plane.
z = 12; 144π ≈ 452.39 sq. units
y = 2; 4π ≈ 12.57 sq. units
35.
36.
6
y
2
6兹3
z
2兹2
Radius of Surface Volume
sphere (m) area (m 2)
(m 3)
1
4␲
2
16␲
4
}} ␲
3
2
10 }}␲
3
3
36␲
36␲
4
64␲
85 }}␲
5
100␲
166 }}␲
37. CRITICAL THINKING Sketch the intersection of a sphere and a plane that
does not pass through the center of the sphere. If you know the circumference
of the circle formed by the intersection, can you find the surface area of the
sphere? Explain. No; you need to know the distance from the center of the sphere
1
3
2
3
34.
2r
to the plane.
SPHERES IN ARCHITECTURE The spherical building below has a
diameter of 165 feet.
38. 27,225π ≈ 85,530 ft 2;
no; Sample answer:
the surface does not
appear to be smooth,
so it is possible that
the building is not
actually a sphere.
38. Find the surface area of a sphere with a
diameter of 165 feet. Looking at the surface of
the building, do you think its surface area is the
same? Explain.
39. The surface of the building consists of 1000
(nonregular) triangular pyramids. If the lateral
area of each pyramid is about 267.3 square feet,
estimate the actual surface area of the building.
r
about 267,300 ft 2
40. Estimate the volume of the building using the
formula for the volume of a sphere.
about 2,352,071 ft 3
BALL BEARINGS In Exercises 41–43, refer to the description of how ball
bearings are made in Example 4 on page 761.
41–43. Answers are
rounded to 2
decimal places.
41. Find the radius of a steel ball bearing made from a cylindrical slug with a
radius of 3 centimeters and a height of 6 centimeters. 3.43 cm
42. Find the radius of a steel ball bearing made from a cylindrical slug with a
radius of 2.57 centimeters and a height of 4.8 centimeters. 2.88 cm
43. If a steel ball bearing has a radius of 5 centimeters, and the radius of the
cylindrical slug it was made from was 4 centimeters, then what was the
height of the cylindrical slug? 10.42 cm
ADDITIONAL PRACTICE
AND RETEACHING
44.
For Lesson 12.6:
• Practice Levels A, B, and C
(Chapter 12 Resource Book,
p. 86)
• Reteaching with Practice
(Chapter 12 Resource Book,
p. 89)
•
See Lesson 12.6 of the
Personal Student Tutor
For more Mixed Review:
•
Search the Test and Practice
Generator for key words or
specific lessons.
764
COMPOSITION OF ICE CREAM In making ice
cream, a mix of solids, sugar, and water is frozen.
Air bubbles are whipped into the mix as it freezes.
The air bubbles are about 1 ª 10º2 centimeter
in diameter. If one quart, 946.34 cubic centimeters,
of ice cream has about 1.446 ª 109 air bubbles,
what percent of the ice cream is air? (Hint: Start
by finding the volume of one air bubble.)
about 80%
764
Chapter 12 Surface Area and Volume
Air bubble
Test
Preparation
MULTI-STEP PROBLEM Use the solids below.
r
4 ASSESS
r
DAILY HOMEWORK QUIZ
2r
48. The volume of the
cylinder is greater than
the volumes of both
other solids; the volume
of the cylinder is equal
to the sum of the
volumes of the sphere
and the double-cone
shaped solid.
★ Challenge
Transparency Available
1. Find the surface area of a
sphere with a radius of 5 m.
Round the result to two
decimal places. 314.16 m 2
2. Find the volume of a sphere
with a diameter of 9.5 ft.
Round the result to two
decimal places. 448.92 ft 3
3. Find the surface area and volume of a sphere with a radius
of 3 cm. Leave the results in
terms of ␲. 36␲ cm 2, 36␲ cm 3
4. Solve for the variable. Then
find the area of the intersection of the sphere and
the plane.
4
3
45. Write an expression for the volume of the sphere in terms of r. }}πr 3
46. Write an expression for the volume of the cylinder in terms of r. 2πr 3
47. Write an expression for the volume of the solid composed of two cones
in terms of r. }2}πr 3
3
48. Compare the volumes of the cylinder and the cones to the volume of the
sphere. What do you notice?
49. A cone is inscribed in a sphere with a radius of
5 centimeters, as shown. The distance from the
center of the sphere to the center of the base of the
cone is x. Write an expression for the volume of
the cone in terms of x. (Hint: Use the radius of the
sphere as part of the height of the cone.)
EXTRA CHALLENGE
www.mcdougallittell.com
5 cm
π
π
}} (25 º x 2)(x + 5) or }} (125 + 25x º 5x 2 º x 3)
3
3
MIXED REVIEW
5
CLASSIFYING PATTERNS Name the isometries that map the frieze pattern
translation, vertical line reflection,
onto itself. (Review 7.6)
180° rotation, glide reflection
translation
50.
51.
52.
41
x
4; 16␲ ≈ 50.27 sq. units
53.
EXTRA CHALLENGE NOTE
translation
translation, 180° rotation
FINDING AREA In Exercises 54–56, determine whether ¤ABC is similar to
14,027
¤EDC. If so, then find the area of ¤ABC. (Review 8.4, 11.3 for 12.7) yes; }
}≈
180
yes; about 19.43 sq. units
yes; 36 sq. units
77.93 sq. units
54.
55.
56. A
A
B
Challenge problems for
Lesson 12.6 are available in
blackline format in the Chapter 12
Resource Book, p. 93 and at
www.mcdougallittell.com.
B
4
C
7
D
17
ADDITIONAL TEST
PREPARATION
1. WRITING What effect does
doubling the radius of a sphere
have on its volume? Explain.
E
A
E
9
C
8 4.5
13
E
B
9
D
8.3
C
D
It multiplies it by 8. The original
4
volume is }}␲ r 3. The new volume
57. MEASURING CIRCLES The tire at the right has
an outside diameter of 26.5 inches. How many
revolutions does the tire make when traveling
100 feet? (Review 11.4) about 14.4 revolutions
3
4
4
3
3
2. WRITING Is there a sphere
is }}␲ (2r) 3 = 8 ⴢ }}␲ r 3.
26.5 in.
12.6 Surface Area and Volume of Spheres
765
with a volume of 288␲ ft 3 and
a surface area of 256␲ ft 2?
Explain.
See sample answer at left.
Additional Test Preparation Sample answer:
4
3
2. No; solving V = 288␲ = }}␲ r 3 for r gives r = 6 ft.
Solving S = 256␲ = 4␲ r 2 for r gives r = 8 ft. In each
case, there is only one possible radius for a given
volume or surface area. Since these radii do not agree,
they describe two spheres of different sizes.
765