2010 AMC 12A 第 7 题

Question

Logan 正在制作他的城镇的比例模型。该城市的水塔高

40

米，顶部是一个可容纳

100{,}000

升水的球体。Logan 的迷你水塔可容纳

0.1

升水。他应该把水塔做成多少米高？

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?

0.04

\dfrac{0.4}{\pi}

0.4

\dfrac{4}{\pi}

4

体积比例是长度比例的立方。

Volume scale is the cube of length scale

比较

0.1

升和

100000

升，再取立方根。

Compare

0.1

liters to

100000

liters, then take a cube root

模型与原物的体积比为

\dfrac{0.1}{100{,}000}=\dfrac{1}{1{,}000{,}000}

。长度按体积比的立方根缩放，所以模型与原物的高度比为

\dfrac1{100}

。因此模型的高度应为

\dfrac{40}{100}=0.4

米。

所以正确答案是 C。

The model-to-original volume ratio is

\dfrac{0.1}{100{,}000}=\dfrac{1}{1{,}000{,}000}.

Lengths scale by the cube root, so the model-to-original height ratio is

\dfrac1{100}.

Therefore the model should be

\dfrac{40}{100}=0.4

meters tall.

Thus, C is the correct answer.

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