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, 7°. what is the ships horizontal distance from the lighthouse (and the shore)? round your answer to the nearest hundredth of a foot if necessary.

Explanation:

Step1: Use the tangent function

In a right - triangle, \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). Here, \(\theta = 7^{\circ}\), the opposite side is the height of the lighthouse (\(h = 131\) feet), and the adjacent side is the horizontal distance \(x\) (what we want to find). So, \(\tan(7^{\circ})=\frac{131}{x}\).

Step2: Solve for \(x\)

Rearrange the formula to \(x=\frac{131}{\tan(7^{\circ})}\). We know that \(\tan(7^{\circ})\approx0.1228\). Then \(x=\frac{131}{0.1228}\).

Step3: Calculate the value of \(x\)

\(x=\frac{131}{0.1228}\approx1066.77\)

Answer:

\(1066.77\) feet