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?

$1.5$

$3$

$4$

$4.5$

$6$

答案：A

知识点：体积圆锥圆柱

难度评级：1020

小提示：

令圆锥体积等于圆柱中水的体积。

Set the cone volume equal to the cylinder volume

大提示：

圆柱半径是圆锥半径的两倍。

The cylinder radius is twice the cone radius

视频讲解：

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文字解答：

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 .

2021 AMC 10B Spring 第 10 题