Question

a boat is heading towards a lighthouse, where yaritza is watching from a vertical distance of 139 feet above the water. yaritza measures an angle of depression to the boat at point $a$ to be $21^{\circ}$. at some later time, yaritza takes another measurement and finds the angle of depression to the boat (now at point $b$) to be $50^{\circ}$. 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
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Explanation:

Step1: Define variables, use angle equality

Let $h=139$ ft (vertical height). Angle of depression = angle of elevation from boat to Yaritza. For point $A$: $\theta_A=21^\circ$, for point $B$: $\theta_B=50^\circ$.

Step2: Find horizontal distance to A

Use $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$, so horizontal distance to $A$ is $x_A=\frac{h}{\tan\theta_A}$.
$\tan(21^\circ)\approx0.3839$, so $x_A=\frac{139}{0.3839}\approx362.07$ ft

Step3: Find horizontal distance to B

Horizontal distance to $B$ is $x_B=\frac{h}{\tan\theta_B}$.
$\tan(50^\circ)\approx1.1918$, so $x_B=\frac{139}{1.1918}\approx116.63$ ft

Step4: Calculate distance AB

$AB = x_A - x_B$ (boat moves closer to lighthouse)
$AB\approx362.07 - 116.63=245.44$

Answer:

245 feet