The top of a hill observed from the top and bottom of a building of height h is at angles of elevation $\frac{\pi}{6}$ and $\frac{\pi}{3}$ respectively. What is the height of the hill?

• A

  2h
• Correct answer
  B

  3h/2
• C

  h
• D

  h/2

Solution

From question, In $\triangle \mathrm{PQR}$ and $\triangle \mathrm{PST}$. $\begin{aligned} & \mathrm{B}=\frac{\mathrm{H}}{\tan \frac{\pi}{3}}=\frac{\mathrm{H}-h}{\tan \frac{\pi}{6}} \\ \Rightarrow \quad & \frac{\mathrm{H}}{3}=\mathrm{H}-h \\ \Rightarrow & \frac{2}{3} \mathrm{H}=h \\ \Rightarrow & \mathrm{H}=\frac{3}{2} h \end{aligned}$

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