The top of a hill when observed from the top and bottom of a building of height $h$ is at angles of elevation p and q respectively. What is the height of the hill?

• A

  $\frac{h \cot q}{\cot q-\cot p}$
• Correct answer
  B

  $\frac{h \cot p}{\cot p-\cot q}$
• C

  $\frac{2 h \tan p}{\tan p-\tan q}$
• D

  $\frac{2 h \tan q}{\tan q-\tan p}$

Solution

$\begin{aligned} & \text {Let height of hill }=\mathrm{H} \\ & \text {& horizontal distance between buildi } \\ & \tan \mathrm{q}=\frac{\mathrm{H}}{\mathrm{~d}} \Rightarrow \mathrm{~d}=\frac{\mathrm{H}}{\tan \mathrm{q}}=\mathrm{H} \cot \mathrm{q} \\ & \tan \mathrm{p}=\frac{(\mathrm{H}-\mathrm{h})}{\mathrm{d}} \Rightarrow \mathrm{~d}=(\mathrm{H}-\mathrm{h}) \cot \mathrm{p} \\ & \Rightarrow \quad H \cot \mathrm{q}=(\mathrm{H}-\mathrm{h}) \cot \mathrm{p} \\ & H=\frac{h \cot \mathrm{p}}{\cot \mathrm{p}-\cot \mathrm{q}} \end{aligned}$ & horizontal distance between building & hill = d

Hence, the correct answer is option: 'B'.