Question

The lower window of a house is at a height of 2 m above the ground and its upper window is 4 m vertically above the lower window . At certain instant the angles of elevation of a balloon from these windows are observed to be 60⁰ and 30⁰ respectively. Find the height of the balloon above the ground.

Answer 1

Let the window G be 2 m above the ground.
Window A be 4 m above the window G.
Balloon be at point B above the ground.
In ∆ABC,
$$\begin{gathered} \tan 30°=\frac{h}{x} \\ \Rightarrow \frac{1}{\sqrt{3}}=\frac{h}{x} \\ \Rightarrow x=h\sqrt{3}.....\left(\mathrm{i}\right) \end{gathered}$$
In ∆BGD,
$$\begin{gathered} \tan 60°=\frac{BD}{GD} \\ \Rightarrow \sqrt{3}=\frac{h+4}{x} \\ \Rightarrow x=\frac{h+4}{\sqrt{3}}.....\left(\mathrm{ii}\right) \end{gathered}$$
From (i) and (ii) we get,
$$\begin{gathered} h\sqrt{3}=\frac{h+4}{\sqrt{3}} \\ \Rightarrow 3h=h+4 \\ \Rightarrow 2h=4 \\ \Rightarrow h=2 \end{gathered}$$
Hence, the height of the ballon above the ground = 2 + 4 + 2 = 8 m