Question

1. a boat heading out to sea starts out at point a, at a horizontal distance of 1216 feet from a lighthouse/the shore. from that point, the boat’s crew measures the angle of elevation to the lighthouse’s beacon - light to be 11°. at some later time, the crew measures the angle of elevation from point b to be 3°. find the distance from point a to point b. round your answer to the nearest tenth of a foot if necessary.

Explanation:

Step1: Find the height of the lighthouse

Let the height of the lighthouse be $h$. At point $A$, we know the horizontal - distance from the lighthouse is $x = 1216$ feet and the angle of elevation $\theta=11^{\circ}$. Using the tangent function $\tan\theta=\frac{h}{x}$, so $h = 1216\times\tan(11^{\circ})$.
$h = 1216\times0.19438 = 236.266$.

Step2: Find the horizontal distance from point $B$ to the lighthouse

Let the horizontal distance from point $B$ to the lighthouse be $y$. We know the height of the lighthouse $h = 236.266$ feet and the angle of elevation at point $B$ is $\alpha = 3^{\circ}$. Using the tangent function $\tan\alpha=\frac{h}{y}$, so $y=\frac{h}{\tan(3^{\circ})}$.
$y=\frac{236.266}{0.0524}=4508.9$.

Step3: Find the distance from point $A$ to point $B$

The distance from point $A$ to point $B$ is $d=y - 1216$.
$d=4508.9-1216=3292.9$.

Answer:

$3292.9$ feet