AMC12 2004 A

AMC12 2004 A · Q9

AMC12 2004 A · Q9. It mainly tests Fractions, Area & perimeter.

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?

一家公司用圆柱形罐子出售花生酱。市场研究表明，使用更宽的罐子会增加销量。如果罐子直径增加 $25\%$ 而体积不变，高度必须减少百分之多少？

(A) 10 10

(B) 25 25

(D) 50 50

(E) 60 60

Answer

Correct choice: (C)

正确答案：（C）

Solution

When the diameter is increased by $25\%$, it is increased by $\dfrac{5}{4}$, so the area of the base is increased by $\left(\dfrac54\right)^2=\dfrac{25}{16}$. To keep the volume the same, the height must be $\dfrac{1}{\frac{25}{16}}=\dfrac{16}{25}$ of the original height, which is a $36\%$ reduction. $\boxed{\mathrm{(C)}\ 36}$

当直径增加 $25\%$ 时，相当于乘以 $\dfrac{5}{4}$，因此底面积增加为 $\left(\dfrac54\right)^2=\dfrac{25}{16}$ 倍。为保持体积不变，高度必须变为原高度的 $\dfrac{1}{\frac{25}{16}}=\dfrac{16}{25}$，这相当于减少 $36\%$。$\boxed{\mathrm{(C)}\ 36}$

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