Question

A house subtends a right angle at the window of the opposite house and the angle of elevation of the window from the bottom of the first house is 60°. If the distance between the two houses is 6m, then the height of the first house is:

• A. \( 8\sqrt{3} \, {m} \)
• B. \( 6\sqrt{3} \, {m} \)
• C. \( 4\sqrt{3} \, {m} \)
• D. None of these

Hint:

In problems involving angles of elevation or depression, use trigonometric ratios such as tangent to relate the height of an object to its distance from the observer.

Answer 1

The Correct Option is A

Let $PQ$ be the house subtending a right angle at the window $B$ of opposite house $AB$. \begin{figure}[ht!] \centering \includegraphics[width=0.5width]{sol125.png} \caption{Diagram of the house and angles} \label{fig:house} \end{figure} In $\triangle ABP$, we have: \[ \tan 60^\circ = \frac{AB}{6} \implies AB = 6\sqrt{3}{m} \] In $\triangle CBQ$, we have: \[ \tan 30^\circ = \frac{h - CP}{BC} \] where $CP = AB$ and $BC = AP$. Since $CP = 6\sqrt{3}{m}$, we have: \[ \frac{1}{\sqrt{3}} = \frac{h - 6\sqrt{3}}{6} \] \[ h - 6\sqrt{3} = 2 \] \[ h = 2 + 6\sqrt{3} = 6\left(\sqrt{3} + \frac{1}{\sqrt{3}}\right) \] \[ h = 8\sqrt{3}{m} \] Hence, the height of the house $h$ is $8\sqrt{3}{m}$. Therefore, the correct answer is Option A.