A sailor judges the distance to a lighthouse by holding a ruler at arm's length and measuring the apparent height of the lighthouse. He knows that the lighthouse is actually 60 feet tall.
If it appears to be 3 inches tall when the ruler is held 2 feet from his eye, how far away is it?
Explanation
The ratio of the ruler's height to the distance from eye to ruler, which is the tangent of the angle subtended at the eye by the ruler's height, must be the same as the ratio of the lighthouse's height to its distance, which is the tangent of the same angle. Since 3 inches is \(\frac{1}{4}\) foot, we have
\(\frac{\frac{1}{4}}{2} = \frac{60}{D}\), and solving for D gives \(D = \frac{(2 \times 60)}{\frac{1}{4}} = 4 \times 120 = 480\) feet.
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