2004 AMC 12A Problem 9

Below is the professionally curated solution for Problem 9 of the 2004 AMC 12A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2004 AMC 12A solutions, or check the answer key.

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

Difficulty rating: 1310

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?

$10$

$25$

$36$

$50$

$60$

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.

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