2004 AMC 12A Problems/Problem 9

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Problem

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?

$\mathrm{(A) \ } 10 \qquad \mathrm{(B) \ } 25 \qquad \mathrm{(C) \ } 36 \qquad \mathrm{(D) \ } 50 {2}\qquad \mathrm{(E) \ } 60$

Solution

When the diameter is increased by $25\%$, is is increased by $\frac54$, so the area of the base is increased by $\left(\frac54\right)^2=\frac{25}{16}$.

To keep the volume the same, the height must be $\frac{1}{\frac{25}{16}}=\frac{16}{25}$ of the original height, which is a $36\%$ reduction $\Rightarrow\mathrm{(C)}$.

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