2021 AMC 10B Spring 第 10 题

Question

一个倒置的圆锥底面半径为

12 \mathrm{ cm}

，高为

18 \mathrm{ cm}

，里面装满了水。把水倒入一个高圆柱中，圆柱的水平底面半径为

24 \mathrm{ cm}

。圆柱中水的高度是多少厘米？

An inverted cone with base radius

12 \mathrm{ cm}

and height

18 \mathrm{ cm}

is full of water. The water is poured into a tall cylinder whose horizontal base has a radius of

24 \mathrm{ cm}.

What is the height in centimeters of the water in the cylinder?

1.5

3

4

4.5

6

令圆锥体积等于圆柱中水的体积。

Set the cone volume equal to the cylinder volume

圆柱半径是圆锥半径的两倍。

The cylinder radius is twice the cone radius

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因为水是从一个容器倒入另一个容器，所以体积保持不变。圆锥中水的体积为

\frac{r^2h \pi}3 = \frac{12^2\cdot 18 \pi}3 = 864 \pi

。

圆柱中水的体积为

r^2h \pi =24^2h \pi = 576h \pi

。于是

576 h \pi = 864 \pi

，所以

h = 1.5

。

所以答案是 A。

The volumes must be the same since the water is poured from one to another. The volume of the cone is

\frac{r^2h \pi}3 = \frac{12^2\cdot 18 \pi}3 = 864 \pi.

The volume of the cylinder is

r^2h \pi =24^2h \pi = 576h \pi.

This makes

576 h \pi = 864 \pi,

so

h = 1.5.

Thus, the answer is A .

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