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