2021 AMC 12B Spring 第 6 题

Question

一个倒置圆锥的底面半径为

12

厘米，高为

18

厘米，里面装满了水。把水倒入一个高圆柱中，这个圆柱的水平底面半径为

24

厘米。圆柱中水的高度是多少厘米？

An inverted cone with base radius

12

cm and height

18

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

24

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

1.5

3

4

4.5

6

圆锥体积是

\dfrac13\pi r^2 h

，圆柱体积是

\pi r^2 h

The volume of a cone is

\dfrac13\pi r^2 h

and of a cylinder is

\pi r^2 h

令圆锥体积等于

\pi\cdot 24^2\cdot h

，并解出

h

。

Set the cone’s volume equal to

\pi\cdot 24^2\cdot h

and solve for

h

圆锥中的水量为

\dfrac13\pi(12)^2(18)=864\pi

立方厘米。

倒入圆柱后，水高为

h

，满足

\pi(24)^2 h=864\pi

。

于是

576h=864

，所以

h=1.5

。

所以正确答案是 A。

The cone holds

\dfrac13\pi(12)^2(18)=864\pi

cubic centimeters of water.

Poured into the cylinder, this fills to height

h

where

\pi(24)^2 h=864\pi.

Then

576h=864,

so

h=1.5.

Thus, the correct answer is A.

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