Question

a boat heading out to sea starts out at point $a$, at a horizontal distance of 1357 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 $9^{\circ}$. at some later time, the crew measures the angle of elevation from point $b$ to be $3^{\circ}$. find the distance from point $a$ to point $b$. round your answer to the nearest tenth of a foot if necessary.

Explanation:

Step1: Find lighthouse height $h$

Let $h$ = height of lighthouse. Use $\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}$ for Point A:
$\tan(9^\circ)=\frac{h}{1357}$
$h = 1357 \times \tan(9^\circ)$
Calculate $\tan(9^\circ)\approx0.1584$, so $h\approx1357\times0.1584\approx214.9$ feet.

Step2: Find distance from shore to Point B

Let $x$ = horizontal distance from shore to Point B. Use $\tan(3^\circ)=\frac{h}{x}$:
$x=\frac{h}{\tan(3^\circ)}$
Calculate $\tan(3^\circ)\approx0.0524$, so $x\approx\frac{214.9}{0.0524}\approx4101.1$ feet.

Step3: Find distance $AB$

Subtract shore distance of A from B:
$AB = x - 1357$
$AB\approx4101.1 - 1357 = 2744.1$ feet.

Answer:

2744.1 feet