AMC12 2010 A

AMC12 2010 A · Q7

AMC12 2010 A · Q7. It mainly tests Ratios & proportions, 3D geometry (volume).

Logan is constructing a scaled model of his town. The city's water tower stands 40 meters high, and the top portion is a sphere that holds 100,000 liters of water. Logan's miniature water tower holds 0.1 liters. How tall, in meters, should Logan make his tower?

Logan 正在制作他城镇的缩比模型。城市的水塔高 40 米，顶部是一个容纳 100,000 升水的球体。Logan 的迷你水塔容纳 0.1 升水。Logan 应该把他的水塔做多高（米）？

(A) 0.04 0.04

(B) \frac{0.4}{\pi} \frac{0.4}{\pi}

(D) \frac{4}{\pi} \frac{4}{\pi}

(E) 4 4

Answer

Correct choice: (C)

正确答案：（C）

Solution

The water tower holds $\frac{100000}{0.1} = 1000000$ times more water than Logan's miniature. The volume of a sphere is: $V=\dfrac{4}{3}\pi r^3$. Since we are comparing the heights (m), we should compare the radii (m) to find the ratio. Since, the radius is cubed, Logan should make his tower $\sqrt[3]{1000000} = 100$ times shorter than the actual tower. This is $\frac{40}{100} = \boxed{0.4}$ meters high, or choice $\textbf{(C)}$. Note: The fact that $1\text{ L}=1000\text{ cm}^3$ doesn't matter since only the ratios are important.

水塔的容水量是 Logan 迷你水塔的 $\frac{100000}{0.1}=1000000$ 倍。球的体积为 $V=\dfrac{4}{3}\pi r^3$。由于要比较高度（米），应比较半径（米）来得到比例。因为半径是三次方关系，Logan 的水塔应比真实水塔短 $\sqrt[3]{1000000}=100$ 倍。因此高度为 $\frac{40}{100}=\boxed{0.4}$ 米，对应选项 $\textbf{(C)}$。注：$1\text{ L}=1000\text{ cm}^3$ 这一事实并不重要，因为只需要用到比值。

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