Solve the problem.
-A person is watching a boat from the top of a lighthouse. The boat is approaching the lighthouse directly. When first noticed, the angle of depression to the boat is 13° 12´. When the boat stops, the
Angle of depression is 49° 49´. The lighthouse is 200 feet tall. How far did the boat travel from when
It was first noticed until it stopped? Round to the nearest foot.
A)706 ft
B)668 ft
C)729 ft
D)684 ft

Question

Quizplus · Accepted Answer

To solve this problem, we use trigonometry, specifically the tangent of an angle in a right triangle, which is the ratio of the opposite side (height of the lighthouse) to the adjacent side (distance from the lighthouse). The angle of depression from the top of the lighthouse to the boat is the same as the angle of elevation from the boat to the top of the lighthouse due to alternate interior angles being congruent. First, we convert the angles from degrees and minutes to decimal degrees for ease of calculation:- 13° 12' = 13 + (12/60) = 13.2°- 49° 49' = 49 + (49/60) = 49.8167°Let $d_1$ be the distance of the boat from the lighthouse when it was first noticed, and $d_2$ be the distance when it stopped. The height of the lighthouse, $h$, is 200 feet.Using the tangent function:1. $\tan(13.2°) = \frac{200}{d_1}$2. $\tan(49.8167°) = \frac{200}{d_2}$Solving each equation for $d_1$ and $d_2$ gives:1. $d_1 = \frac{200}{\tan(13.2°)}$2. $d_2 = \frac{200}{\tan(49.8167°)}$Calculating these distances:1. $d_1 ≈ 844.09$ feet2. $d_2 ≈ 160.09$ feetThe distance the boat traveled is the difference between these two distances:$d_1 - d_2 ≈ 844.09 - 160.09 ≈ 684$ feet, rounded to the nearest foot.Therefore, the boat traveled approximately 684 feet from when it was first noticed until it stopped.

Solve the Problem. -A Person Is Watching a Boat from the Top of the Top