Question

a boat is heading towards a lighthouse, where jeriel is watching from a vertical distance of 113 feet above the water. jeriel measures an angle of depression to the boat at point a to be 8°. at some later time, jeriel takes another measurement and finds the angle of depression to the boat (now at point b) to be 52°. find the distance from point a to point b. round your answer to the nearest tenth of a foot if necessary.

Explanation:

Step1: Define known values

Vertical height $h = 113$ ft, $\angle A = 8^\circ$, $\angle B = 52^\circ$

Step2: Find distance from base to A

Use $\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}$, so distance to A: $d_A = \frac{h}{\tan(8^\circ)}$
$\tan(8^\circ)\approx0.1405$, so $d_A=\frac{113}{0.1405}\approx804.27$ ft

Step3: Find distance from base to B

Distance to B: $d_B = \frac{h}{\tan(52^\circ)}$
$\tan(52^\circ)\approx1.2799$, so $d_B=\frac{113}{1.2799}\approx88.30$ ft

Step4: Calculate distance A to B

Subtract $d_B$ from $d_A$: $AB = d_A - d_B$

Answer:

$804.27 - 88.30 = 715.97 \approx 716.0$ feet