Question

a boat is heading towards a lighthouse, where jeremiah is watching from a vertical distance of 126 feet above the water. jeremiah measures an angle of depression to the boat at point $a$ to be $11^{\circ}$. at some later time, jeremiah takes another measurement and finds the angle of depression to the boat (now at point $b$) to be $78^{\circ}$. 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: Define right triangles for points A/B

We have two right triangles: vertical side = 126 ft, angles of depression equal to the angles at the boat (alternate interior angles). For point A: angle = $11^\circ$, for point B: angle = $78^\circ$.

Step2: Calculate distance from base to A

Use tangent: $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$, so distance to A is $\frac{126}{\tan(11^\circ)}$
$\frac{126}{\tan(11^\circ)} \approx \frac{126}{0.1944} \approx 648.15$ ft

Step3: Calculate distance from base to B

Distance to B is $\frac{126}{\tan(78^\circ)}$
$\frac{126}{\tan(78^\circ)} \approx \frac{126}{4.7046} \approx 26.78$ ft

Step4: Find distance A to B

Subtract the two distances: $648.15 - 26.78$

Answer:

621.4 feet