AMC12 2021 B

AMC12 2021 B · Q6

AMC12 2021 B · Q6. It mainly tests Fractions, 3D geometry (volume).

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 radius of $24 \mathrm{cm}$. What is the height in centimeters of the water in the cylinder?

一个底面半径为$12 \mathrm{cm}$、高$18 \mathrm{cm}$的倒锥体装满了水。水被倒入一个高圆柱体中，该圆柱体的水平底面半径为$24 \mathrm{cm}$。圆柱体中水的液面高度是多少厘米？

(A) 1.5 1.5

(B) 3 3

(D) 4.5 4.5

(E) 6 6

Answer

Correct choice: (A)

正确答案：（A）

Solution

The volume of a cone is $\frac{1}{3} \cdot\pi \cdot r^2 \cdot h$ where $r$ is the base radius and $h$ is the height. The water completely fills up the cone so the volume of the water is $\frac{1}{3}\cdot18\cdot144\pi = 6\cdot144\pi$. The volume of a cylinder is $\pi \cdot r^2 \cdot h$ so the volume of the water in the cylinder would be $24\cdot24\cdot\pi\cdot h$. We can equate these two expressions because the water volume stays the same like this $24\cdot24\cdot\pi\cdot h = 6\cdot144\pi$. We get $4h = 6$ and $h=\frac{6}{4}$. So the answer is $\boxed{\textbf{(A)} ~1.5}.$

圆锥的体积公式为$\frac{1}{3} \cdot\pi \cdot r^2 \cdot h$，其中$r$是底面半径，$h$是高度。水完全充满锥体，因此水的体积为$\frac{1}{3}\cdot18\cdot144\pi = 6\cdot144\pi$。圆柱的体积为$\pi \cdot r^2 \cdot h$，因此圆柱中水的体积为$24\cdot24\cdot\pi\cdot h$。由于水的体积不变，我们可以设等式$24\cdot24\cdot\pi\cdot h = 6\cdot144\pi$。得到$4h = 6$，$h=\frac{6}{4}$。因此答案为$\boxed{\textbf{(A)} ~1.5}$。

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