2010 AMC 12A Problem 7

Question

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

Volume scale is the cube of length scale

Compare

0.1

liters to

100000

liters, then take a cube root

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.

Problem 7 in other years