Question

a boat is heading towards a lighthouse, whose beacon - light is 131 feet above the water. the boats crew measures the angle of elevation to the beacon, 4°. what is the ships horizontal distance from the lighthouse (and the shore)? round your answer to the nearest tenth of a foot if necessary.

Explanation:

Step1: Use the tangent function

In a right - triangle, \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). Here, \(\theta = 4^{\circ}\), the opposite side is the height of the lighthouse (\(h = 131\) feet), and the adjacent side is the horizontal distance \(x\) we want to find. So, \(\tan(4^{\circ})=\frac{131}{x}\).

Step2: Solve for \(x\)

We can rewrite the equation as \(x=\frac{131}{\tan(4^{\circ})}\).
We know that \(\tan(4^{\circ})\approx0.0699\). Then \(x=\frac{131}{0.0699}\approx1874.1\).

Answer:

\(1874.1\) feet