Question

a boat is heading towards a lighthouse, whose beacon - light is 132 feet above the water. the boats crew measures the angle of elevation to the beacon, ( 5^{circ} ). 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, \(\theta = 5^{\circ}\), the opposite side is the height of the lighthouse (\(h = 132\) feet), and the adjacent side is the horizontal distance \(x\) (which we want to find). So, \(\tan(5^{\circ})=\frac{132}{x}\).

Step2: Solve for \(x\)

We can rewrite the equation as \(x=\frac{132}{\tan(5^{\circ})}\).
We know that \(\tan(5^{\circ})\approx0.08748877\).
Then \(x=\frac{132}{0.08748877}\).
\(x\approx1508.79\)

Answer:

The ship's horizontal distance from the lighthouse (and the shore) is approximately \(1508.79\) feet.