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

To find the distance to the lighthouse, the sailor can use similar triangles and set up a proportion between the actual height of the lighthouse and the apparent height observed. By plugging in the values and solving the equation, the sailor can determine that the lighthouse is 40 feet away.

Step-by-step explanation:

To solve this problem, we can use similar triangles to find the distance to the lighthouse. We can set up a proportion between the actual height of the lighthouse and the apparent height observed by the sailor. Since the ruler is held at arm's length, which is 2 feet, and it appears to be 3 inches tall, we can set up the proportion as:

Actual height of lighthouse / Apparent height of lighthouse = Distance to lighthouse / Arm's length

Plugging in the values, we get:

60 feet / 3 inches = Distance to lighthouse / 2 feet

Cross-multiplying, we get:

3 inches * Distance to lighthouse = 2 feet * 60 feet

Converting the units, we have:

3 inches * Distance to lighthouse = 120 feet

Dividing both sides by 3 inches, we find:

Distance to lighthouse = 40 feet

Therefore, the lighthouse is 40 feet away from the sailor.

Answer 2

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