Two ships are sailing in the sea on either side of a lighthouse. The angles of depression to the two ships as observed from the top of the lighthouse are $60°$ and $45°$, respectively. If the distance

Question

Two ships are sailing in the sea on either side of a lighthouse. The angles of depression to the two ships as observed from the top of the lighthouse are $60°$ and $45°$, respectively. If the distance between the ships is $100\left(\frac{1 + \sqrt{3}}{\sqrt{3}}\right)$ m, then find the height of the lighthouse.

Accepted Answer

Step 1: Lighthouse AB with ships at P and Q, angles of depression 60° and 45°
Step 2: Let $AB$ be the height of the lighthouse. In right $	riangle ABP$:
$\frac{AB}{PB} = 	an 60° = \sqrt{3}$
$\Rightarrow PB = \frac{AB}{\sqrt{3}}$ ... (1)
Step 3: In right $	riangle ABQ$:
$\frac{AB}{BQ} = 	an 45° = 1$
$\Rightarrow BQ = AB$ ... (2)
Step 4: Adding (1) and (2), we have
$PB + BQ = \frac{AB}{\sqrt{3}} + AB$
$\Rightarrow PQ = AB\left(\frac{1 + \sqrt{3}}{\sqrt{3}}ight)$
Step 5: $\Rightarrow 100\left(\frac{1 + \sqrt{3}}{\sqrt{3}}ight) = AB\left(\frac{1 + \sqrt{3}}{\sqrt{3}}ight)$
$\Rightarrow AB = 100$ m

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