Question

a boat is heading towards a lighthouse, whose beacon - light is 111 feet above the water. the boats crew measures the angle of elevation to the beacon, 13°. 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, \(\theta = 13^{\circ}\), the opposite side is \(x\) (height of the lighthouse above water) and the adjacent side is the horizontal distance from the boat to the shore. We know that \(\tan\theta=\frac{x}{111}\).

Step2: Solve for \(x\)

Since \(\tan(13^{\circ})\approx0.2309\) (using a calculator), and \(\tan(13^{\circ})=\frac{x}{111}\), we can solve for \(x\) by multiplying both sides of the equation by \(111\). So \(x = 111\times\tan(13^{\circ})\).
Substitute \(\tan(13^{\circ})\approx0.2309\) into the equation: \(x=111\times0.2309 = 25.6299\approx25.63\)

Answer:

\(25.63\)