Question

a boat heading out to sea starts out at point $a$, at a horizontal distance of 1115 feet from a lighthouse/the shore. from that point, the boats crew measures the angle of elevation to the lighthouses beacon-light to be $7^{\circ}$. at some later time, the crew measures the angle of elevation from point $b$ to be $2^{\circ}$. find the distance from point $a$ to point $b$. round your answer to the nearest tenth of a foot if necessary.

Explanation:

Step1: Find height of lighthouse

Let $h$ = height of lighthouse. Use $\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}$ for point A.
$\tan(7^\circ)=\frac{h}{1115}$
$h = 1115 \times \tan(7^\circ)$
$h \approx 1115 \times 0.1228 = 136.922$ feet

Step2: Find distance from B to L

Let $x$ = distance from B to L. Use $\tan(2^\circ)=\frac{h}{x}$.
$x = \frac{h}{\tan(2^\circ)}$
$x \approx \frac{136.922}{0.0349} \approx 3923.266$ feet

Step3: Calculate distance AB

$AB = x - 1115$
$AB \approx 3923.266 - 1115$

Answer:

2808.3 feet