Final answer:
To calculate the distance the boat traveled, trigonometry was used with the angles of depression and the height of the lighthouse. The boat traveled approximately 559.69 feet from when it was first noticed until it stopped.
Step-by-step explanation:
To solve this problem, we need to use trigonometry to find the distance the boat traveled while approaching the lighthouse.
First, we calculate the distance of the boat from the lighthouse when it was first noticed using the initial angle of depression. Given the height of the lighthouse (200 feet) and the angle of depression (14°52'), we can use the tangent function:
Tan(14°52') = distance/200ft
We solve for distance to get the initial distance the boat is from the lighthouse. Then, we do the same for the final position of the boat using the angle of depression when the boat stopped (45°10').
Tan(45°10') = distance/200ft
The difference between these two distances gives us the distance the boat traveled.
Applying the tangent function to the angles and solving for the distances using a calculator, we get approximately 759.69 feet for the initial angle and precisely 200 feet for the final angle due to the tangent of 45° being 1. So, the boat traveled approximately the difference between these two distances.
The boat traveled approximately 759.69ft - 200ft which gives us 559.69ft.