Question

a boat heading out to sea starts out at point $a$, at a horizontal distance of 1465 feet from a lighthouse/the shore. from that point, the boats crew measures the angle of elevation to the lighthouses beacon-light from that point to be $13^{\circ}$. at some later time, the crew measures the angle of elevation from point $b$ to be $8^{\circ}$. find the distance from point $a$ to point $b$. round your answer to the nearest foot if necessary.

Explanation:

Step1: Find lighthouse height $h$

Let $h$ = height of lighthouse. Use $\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}$ for Point A:
$\tan(13^\circ)=\frac{h}{1465}$
$h = 1465 \times \tan(13^\circ)$
Calculate $\tan(13^\circ)\approx0.2309$, so $h\approx1465\times0.2309\approx338.27$ feet

Step2: Find distance from shore to B

Let $x$ = horizontal distance from shore to Point B. Use $\tan(8^\circ)=\frac{h}{x}$:
$x=\frac{h}{\tan(8^\circ)}$
Calculate $\tan(8^\circ)\approx0.1405$, so $x\approx\frac{338.27}{0.1405}\approx2407.62$ feet

Step3: Find distance $AB$

$AB = x - 1465$
$AB\approx2407.62 - 1465 = 942.62$

Answer:

943 feet (rounded to nearest foot)