2010 AMC 12A Problems/Problem 7

Problem

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?

$\textbf{(A)}\ 0.04 \qquad \textbf{(B)}\ \frac{0.4}{\pi} \qquad \textbf{(C)}\ 0.4 \qquad \textbf{(D)}\ \frac{4}{\pi} \qquad \textbf{(E)}\ 4$

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.

Video Solution by the Beauty of Math

https://youtu.be/kU70k1-ONgM?t=154

2010 AMC 12A (Problems • Answer Key • Resources)
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All AMC 12 Problems and Solutions

2010 AMC 10A (Problems • Answer Key • Resources)
Preceded by Problem 11	Followed by Problem 13
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All AMC 10 Problems and Solutions

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2010 AMC 12A Problems/Problem 7

Contents

Problem

Solution

Video Solution by the Beauty of Math

See also