Let's call the height of the lighthouse "h". In order for the sailor to see the top of the lighthouse in the mirror, the angle of incidence must equal the angle of reflection.
Let's label the point where the sailor's eye is as point A, the top of the lighthouse as point B, the base of the lighthouse as point C, and the point on the ground directly below the top of the lighthouse as point D.
Since the sailor is 4 feet away from the mirror, point A is 4 feet away from the mirror. Since his eyes are 5 feet 6 inches above the ground, point A is 5.5 feet above the ground. Thus, point A is located at (4, 5.5).
Since the mirror is 10 feet from the lighthouse, point D is 10 feet from the lighthouse. Since the sailor's eye is 5 feet 6 inches above the ground, point D is located at (0, 5.5).
We can create a right triangle with points A, D, and B. The distance from point A to point D is 4 feet, the distance from point D to point B is h feet (the height of the lighthouse), and the distance from point A to point B is 10 feet.
Using the Pythagorean theorem, we have:
(4)^2 + (h)^2 = (10)^2
16 + h^2 = 100
h^2 = 84
h = √84
h ≈ 9.2 feet
Therefore, the height of the lighthouse is approximately 9.2 feet.
A sailor is standing next to a lighthouse. He places a mirror on the ground 10 feet from the lighthouse and walks backward until he can see the top in the mirror. He is 4 feet away from the mirror, and his eyes are 5 feet 6 inches above the ground. What is the height of the lighthouse?
1 answer