Question

a boat is spotted in the water with an angle of depression of 25° from the top of a lighthouse that is 89 feet tall. to the nearest foot, how far away is the boat from the base of the lighthouse? a 37 feet b 42 feet c 188 feet d 191 feet

Explanation:

Step1: Use the tangent function

The angle of depression is \(25^{\circ}\), and the height of the lighthouse (opposite side) is \(h = 89\) feet. Let the distance of the boat from the base of the lighthouse (adjacent side) be \(x\). Since the angle of depression is equal to the angle of elevation from the boat to the top of the lighthouse. We know that \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\), so \(\tan(25^{\circ})=\frac{89}{x}\).

Step2: Solve for \(x\)

Rearrange the formula to \(x=\frac{89}{\tan(25^{\circ})}\). We know that \(\tan(25^{\circ})\approx0.4663\). Then \(x = \frac{89}{0.4663}\approx191\)

Answer:

D. 191 feet