2010 AMC 12A Problem 7

Below is the professionally curated solution for Problem 7 of the 2010 AMC 12A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2010 AMC 12A solutions, or check the answer key.

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Concepts:power scaling of length, area, and volume volume

Difficulty rating: 1420

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?

$0.04$

$\dfrac{0.4}{\pi}$

$0.4$

$\dfrac{4}{\pi}$

$4$

Solution:

The miniature tower holds $\dfrac{100,000}{.1} = 1,000,000$ times less water than the actual tower. Since this is the ratio for volumes, the ratio of heights is $(1,000,000)^{1 / 3} = 100.$ This means that the height of the miniature tower is $\dfrac{40}{100} = .4.$

Thus, C is the correct answer.

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