Question

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question
a boat is heading towards a lighthouse, where luis is watching from a vertical distance of 149 feet above the water. luis measures an angle of depression to the boat at point a to be $14^\circ$. at some later time, luis takes another measurement and finds the angle of depression to the boat (now at point b) to be $56^\circ$. find the distance from point a to point b. round your answer to the nearest tenth of a foot if necessary.
answer attempt 1 out of 2
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Explanation:

Step1: Find distance from L to A

Let $d_{LA}$ = horizontal distance from lighthouse (L) to point A. Use $\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}$, where $\theta=14^\circ$, opposite side = 149 ft.
$\tan(14^\circ)=\frac{149}{d_{LA}}$
$d_{LA}=\frac{149}{\tan(14^\circ)}$
$d_{LA}\approx\frac{149}{0.2493}\approx597.67$ ft

Step2: Find distance from L to B

Let $d_{LB}$ = horizontal distance from lighthouse (L) to point B. Use $\tan(56^\circ)=\frac{149}{d_{LB}}$.
$d_{LB}=\frac{149}{\tan(56^\circ)}$
$d_{LB}\approx\frac{149}{1.4826}\approx100.50$ ft

Step3: Calculate distance A to B

Subtract $d_{LB}$ from $d_{LA}$: $d_{AB}=d_{LA}-d_{LB}$
$d_{AB}\approx597.67 - 100.50$

Answer:

497.2 feet