Learn to analyze and compare the effects of changing dimensions of 3dimensional figures and how to c

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Learn to analyze and compare the effects of
changing dimensions of 3-dimensional figures and
how to compare unlike figures.
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Additional Example Exploring the Effects of
Changing Dimensions
A cylinder has diameter 8 in. and height 3 in.
Explain whether tripling the height would have
the same effect on the surface area as tripling
the radius.
They would not have the same effect. Tripling the
radius would increase the surface area more than
tripling the height.
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Try This Example 2
A cylinder has diameter 6 in. and height 2 in.
Explain whether doubling the height would have
the same effect on the surface area as doubling
the radius.
S 2pr2 2pr(2h)
S 2pr2 2p(2r)h
S 2pr² 2prh
2p(3)2 2p(3)(4)
2p(6) 2 2p(3)(2)
2p(3)2 2p(3)(2)
42p in2 131.88 in2
84p in2 263.76 in2
30p in2 94.2 in2
They would not have the same effect. Doubling the
radius would increase the surface area more than
doubling the height.
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Try This Example
A cone has diameter 9 in. and a slant height 2
in. Explain whether tripling the slant height
would have the same effect on the surface area as
tripling the radius.
S pr2 pr(3l)
S pr2 prl
S p(3r)2 p(3r)l
p(4.5)2 p(4.5)(2)
p(4.5)2 p(4.5)(6)
p(13.5)2 p(13.5)(2)
29.25p in2 ? 91.8 in2
47.25p in2 ? 148.4 in2
209.25p in2 ? 657.0 in2
They would not have the same effect. Tripling the
radius would increase the surface area more than
tripling the height.
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Additional Example Application
The upper portion of an hourglass is
approximately an inverted cone with the given
dimensions. What is the lateral surface area of
the upper portion of the hourglass?
Pythagorean Theorem
a2 b2 l2
102 262 l2
l ? 27.9
Lateral surface area
L prl
p(10)(27.9) ? 876.1 mm2
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Lesson Quiz
3. Tell whether doubling the dimensions of a
cone will double the surface area.
It will more than double the surface area because
you square the radius to find the area of the
base.
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Additional Example 3 Comparing Volumes and
Surface Areas
Compare the volumes and surface areas of a sphere
with radius 42 cm with that of a rectangular
prism measuring 44 cm ? 84 cm ? 84 cm.
Sphere
Rectangular Prism
V lwh
(44)(84)(84)
? 310,464 cm3
310,464 cm3
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Additional Example 3 Continued
Compare the volumes and surface areas of a sphere
with radius 42 cm with that of a rectangular
prism measuring 44 cm ? 84 cm ? 84 cm.
Sphere
Rectangular Prism
S 4pr2 4p(422)
S 2lw 2lh 2wh
7,056p
28,896 cm2
The sphere and the prism have approximately the
same volume, but the prism has a larger surface
area.
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Try This Example 3
Compare the volume and surface area of a sphere
with radius 21 mm with that of a rectangular
prism measuring 22 ? 42 ? 42 mm.
Sphere
Rectangular Prism
V lwh
(22)(42)(42)
? 38,808 mm3
38,808 mm3
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Try This Example 3 Continued
Compare the volume and surface area of a sphere
with radius 21 mm with that of a rectangular
prism measuring 22 ? 42 ? 42 mm.
Sphere
Rectangular Prism
S 4pr2 4p(212)
S 2lw 2lh 2wh
1764p
7224 mm2
The sphere and the prism have approximately the
same volume, but the prism has a larger surface
area.
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Lesson Quiz Part 1
Find the volume of each sphere, both in terms of
? and to the nearest tenth. Use 3.14 for p. 1. r
4 ft 2. d 6 m Find the surface area of each
sphere, both in terms of ? and to the nearest
tenth. Use 3.14 for p.
85.3p ft3, 267.8 ft3
36p m3, 113.0 m3
1936p in2, 6079.0 in2
3. r 22 in
4. d 1.5 mi
2.25p mi2, 7.1 mi2
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Lesson Quiz Part 2
5. A basketball has a circumference of 29 in. To
the nearest cubic inch, what is its volume?
412 in3