Question

A lighthouse keeper, Aaden, observes a boat from two different locations. Initially, Aaden measures the angle of depression from Point A to the boat as 18 degrees. At a later time, from Point B, the angle of depression to the same boat becomes 75 degrees. If Aaden is 103 feet above the water level, what is the distance between Point A and Point B to the nearest tenth of a foot?

A) 206.2 feet
B) 172.3 feet
C) 98.6 feet
D) 124.8 feet

Answer 1

To find the distance between Point A and Point B, we can use the concept of trigonometry. We create two right triangles, one with an angle of depression of 18 degrees and the other with an angle of depression of 75 degrees. By using the tangent function, we can solve for the distances x and y and then find the difference between them to get the distance between Point A and Point B.

Step-by-step explanation:

To find the distance between Point A and Point B, we can use the concept of trigonometry. Let's denote the distance between Point A and the boat as x, and the distance between Point B and the boat as y. We can create two right triangles, one with an angle of depression of 18 degrees and the other with an angle of depression of 75 degrees.

In the first triangle, the side opposite the angle of depression is x, and the side adjacent to the angle is the height of the lighthouse, which is 103 feet. Using the tangent function, we can write:

tan(18) = x/103

Solving for x, we find that x ≈ 37.6 feet.

In the second triangle, the side opposite the angle of depression is y, and the side adjacent to the angle is the height of the lighthouse. Using the tangent function again, we can write:

tan(75) = y/103

Solving for y, we find that y ≈ 206.2 feet.

Finally, to find the distance between Point A and Point B, we can subtract the distances x and y:

Distance between Point A and Point B = y - x = 206.2 - 37.6 ≈ 168.6 feet