2004 AMC 12A Problem 9

Question

A company sells peanut butter in cylindrical jars. Marketing research suggests that using wider jars will increase sales. If the diameter of the jars is increased by

25\%

without altering the volume, by what percent must the height be decreased?

10

25

36

50

60

Answer: C

Concepts:cylinder power scaling of length, area, and volume percentage

Difficulty rating: 1310

Small hint:

The base area scales with the square of the diameter

Big hint:

Multiplying the diameter by

\tfrac54

multiplies the base area by

\left(\tfrac54\right)^2

Solution:

Multiplying the diameter by

\tfrac54

multiplies the base area by

\left(\tfrac54\right)^2 = \tfrac{25}{16}.

To keep the volume fixed, the height must be multiplied by

\tfrac{16}{25} = 0.64.

That is a decrease of

1 - 0.64 = 0.36,

or

36\%.

Thus, the correct answer is C.