43
Example: scaling of the volume of a sphere
Question: In figure (c), the larger sphere has a radius that is five
times greater. How many times greater is its volume.
Correct solution #1: Volume scales like the third power of the
linear size, so the larger sphere has a volume that is 125 times
greater (5
3
=125).
Correct solution #2: The volume of a sphere is V=
4
3
p
r
3
, so
V
1
=
4
3
p
r
1
3
V
2
=
4
3
p
r
2
3
=
4
3
p
(5r
1
)
3
=
500
3
p
r
1
3
V
2
/V
1
=
500
3
p
r
1
3
/4
3
p
r
1
3
= 125
Incorrect solution: The volume of a sphere is V=
4
3
p
r
3
, so
V
1
=
4
3
p
r
1
3
V
2
=
4
3
p
r
2
3
=
4
3
p·
5r
1
3
=
20
3
p
r
1
3
V
2
/V
1
=(
20
3
p
r
1
3
)/(
4
3
p
r
1
3
)
=5
(The solution is incorrect because (5r
1
)
3
is not the same as
5r
1
3
.)
Example: scaling of a more complex shape
Question: The first letter “S” in fig. (d) is in a 36-point font, the
second in 48-point. How many times more ink is required to make
the larger “S”.
Correct solution: The amount of ink depends on the area to be
covered with ink, and area is proportional to the square of the linear
dimensions, so the amount of ink required for the second “S” is
greater by a factor of (48/36)
2
=1.78.
Incorrect solution: The length of the curve of the second “S” is
longer by a factor of 48/36=1.33, so 1.33 times more ink is
required.
(The solution is wrong because it assumes incorrectly that the width
of the curve is the same in both cases. Actually both the width and
the length of the curve are greater by a factor of 48/36, so the area is
greater by a factor of (48/36)
2
=1.78.)
S
S
(d) The 48-point “S” has 1.78 times
more area than the 36-point “S.”
(c)
The big sphere has 125 times more
volume than the little one.
Section 1.2Scaling of Area and Volume