Question

A boat is heading towards a lighthouse, whose beacon-light is 140 feet above the water. From point AA, the boat’s crew measures the angle of elevation to the beacon, 13degrees ∘ , before they draw closer. They measure the angle of elevation a second time from point BB at some later time to be 25degrees ∘ . Find the distance from point AA to point BB. Round your answer to the nearest foot if necessary.

Answer 1

The distance from point AA to point BB is found using trigonometry by calculating the individual distances from each point to the lighthouse and then subtracting one from the other.

Step-by-step explanation:

To find the distance from point AA to point BB, we can use trigonometry and the concept of right-angled triangles formed by the angle of elevation from each point to the lighthouse beacon. Let's denote the distance from AA to the lighthouse as d1, and from BB to the lighthouse as d2. The distance between AA and BB will be d1 - d2.

At point AA, the angle of elevation is 13° and the height of the lighthouse is 140 feet. Using the tangent function, we have:

1. tan(13°) = 140 / d1
   
   Solving for d1 gives us: d1 = 140 / tan(13°)

2. 
   Similarly, at point BB, where the angle of elevation is 25°:

After calculating the distances d1 and d2, subtract d2 from d1 to find the distance between AA and BB. Round the result to the nearest foot if necessary.