AMC10 2004 A

AMC10 2004 A · Q11

AMC10 2004 A · Q11. 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。如果罐子的直径增加25%，体积不变，则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

(C) Let $r$, $h$, and $V$, respectively, be the radius, height, and volume of the jar that is currently being used. The new jar will have a radius of $1.25r$ and volume $V$. Let $H$ be the height of the new jar. Then $\pi r^2 h = V = \pi(1.25r)^2H$, so $\dfrac{H}{h}=\dfrac{1}{(1.25)^2}=0.64.$ Thus $H$ is $64\%$ of $h$, so the height must be reduced by $(100-64)\%=36\%$. OR Multiplying the diameter by $5/4$ multiplies the area of the base by $(5/4)^2=25/16$, so in order to keep the same volume, the height must be multiplied by $16/25$. Thus the height must be decreased by $9/25$, or $36\%$.

（C）设当前使用的罐子的半径、高度和体积分别为 $r$、$h$ 和 $V$。新罐子的半径为 $1.25r$，体积仍为 $V$。设新罐子的高度为 $H$。则 $\pi r^2 h = V = \pi(1.25r)^2H$，所以 $\dfrac{H}{h}=\dfrac{1}{(1.25)^2}=0.64.$ 因此 $H$ 是 $h$ 的 $64\%$，所以高度必须减少 $(100-64)\%=36\%$。或者直径乘以 $5/4$ 会使底面积乘以 $(5/4)^2=25/16$，为了保持体积不变，高度必须乘以 $16/25$。因此高度必须减少 $9/25$，即 $36\%$。

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