Question

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?

Answer 1

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 ¼ foot, we have ¼/2=60/D, and solving for D gives D= 2×60/¼ = 4 × 120 = 480 feet

Answer 2

Solution:

Actual Height of Light House = 60 Feet

Comparing with the ruler

Height of Ruler = 3 inches

Distance between eye and Ruler = 2 feet

Let Actual Distance between Light house and Ruler = x feet

As the two triangles are similar by AA Criteria, so their sides  will be proportional to each other, i.e

$\frac{60}{3}=\frac{x}{2}\\\\ 20=\frac{x}{2}\\\\ x=40$ feet

So, Distance between Light house and Ruler = 40 feet