A lighthouse

A lighthouse overlooks a bay, and it is 77 meters high. From the top, the lighthouse keeper can see a yacht southward at an angle of depression of 32 degrees and another boat eastward at an angle of 25 degrees. What is the distance between the boats?

Final Answer:

x =  206 m

Step-by-step explanation:

$$\begin{gathered} h=77\text{m} \\ \alpha =32°\to \text{rad}=32°\cdot \frac{\pi }{180}=32°\cdot \frac{3.1415926}{180}=0.55851=8\pi /45 \\ \beta =25°\to \text{rad}=25°\cdot \frac{\pi }{180}=25°\cdot \frac{3.1415926}{180}=0.43633=5\pi /36 \\ \tan \alpha =h:a=h/a \\ a=h/\tan \alpha =77/\tan 0.5585\doteq 123.2258\text{m} \\ \tan \beta =h:b=h/b \\ b=h/\tan \beta =77/\tan 0.4363\doteq 165.127\text{m} \\ {x}^2=a^2+b^2 \\ x=\sqrt{a^2+b^2}=\sqrt{123.2258^2+165.127^2}=206\text{m} \end{gathered}$$

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