Question

question
a boat heading out to sea starts out at point $a$, at a horizontal distance of 1315 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 $12^{\circ}$. 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.
answer attempt 1 out of 2
feet

Explanation:

Step1: Find height of lighthouse

Let $h$ = height of lighthouse. Use $\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}$ for Point A.
$\tan(12^\circ)=\frac{h}{1315}$
$h = 1315 \times \tan(12^\circ)$
Calculate: $h \approx 1315 \times 0.2126 = 279.569$ feet

Step2: Find distance from shore to Point B

Use the height $h$ to find horizontal distance $d_B$ from shore to B.
$\tan(8^\circ)=\frac{h}{d_B}$
$d_B = \frac{h}{\tan(8^\circ)}$
Calculate: $d_B \approx \frac{279.569}{0.1405} = 1989.815$ feet

Step3: Find distance A to B

Subtract distance from shore to A from distance to B.
$AB = d_B - 1315$
$AB \approx 1989.815 - 1315 = 674.815$

Step4: Round to nearest foot

Round the result to whole number.
$AB \approx 675$

Answer:

675 feet