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: Recall tangent - function formula

We know that in a right - triangle, $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. Here, the height of the lighthouse (opposite side) is $h = 131$ feet and the angle of elevation $\theta=4^{\circ}$, and the horizontal distance $x$ is the adjacent side. So, $\tan\theta=\tan(4^{\circ})=\frac{131}{x}$.

Step2: Solve for $x$

We can rewrite the equation as $x=\frac{131}{\tan(4^{\circ})}$. Since $\tan(4^{\circ})\approx0.0699$, then $x=\frac{131}{0.0699}\approx1874.1$.

Answer:

$1874.1$ feet