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The top of a hill when observed from the top and bottom of a building of height $\mathrm{h}$ is at angles of elevation $\mathrm{p}$ and $\mathrm{q}$ respectively. What is the height of the hill?
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The correct answer is:
$\frac{\text { hcot } p}{\cot p-\cot q}$
Let height of hill $=\mathrm{H}$ \& horizontal distance between building \& hill = d
$\tan \mathrm{q}=\frac{\mathrm{H}}{\mathrm{d}} \Rightarrow \mathrm{d}=\frac{\mathrm{H}}{\tan \mathrm{q}}=\mathrm{H} \cot \mathrm{q}$
$\tan \mathrm{p}=\frac{(\mathrm{H}-\mathrm{h})}{\mathrm{d}} \Rightarrow \mathrm{d}=(\mathrm{H}-\mathrm{h}) \cot \mathrm{p}$
$\Rightarrow \mathrm{H} \cot \mathrm{q}=(\mathrm{H}-\mathrm{h}) \cot \mathrm{p}$
$H=\frac{h \cot p}{\cot p-\cot q}$
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