Question

Suppose, there are two windows in a house. A window of the house is at a height of $1.5m$ above the ground and the other window is $3m$ vertically above the lower window. Anil and Sanjeev are sitting inside the two windows. At an instant, the angles of elevation of a balloon from these windows are observed as ${45}^{\circ }$ and ${30}^{\circ }$, respectively.
Find the height of the balloon from the ground.

Answer 1

Let $GC$ be the height of the balloon from the ground. Let $A$ and $D$ be the positions of two windows such that $AF=BG=1.5m,AD=BE=3m$ where $DE\perp GC$ and $AB\perp GC$.

Let $BC=h$ such that $CE=h-3$.

Consider right $\Delta CAB$

$\tan {45}^{\circ }=\frac{BC}{AB}$

$1=\frac{h}{x}$

$\therefore x=h$ or $AB=BC=DE$...(i)

Consider right $\Delta CDE$

$\tan {30}^{\circ }=\frac{CE}{DE}$

$\frac{1}{\sqrt{3}}=\frac{h-3}{h}$

$h=\sqrt{3}h-3\sqrt{3}$

$h(\sqrt{3}-1)=3\sqrt{3}$

$h=\frac{3\sqrt{3}}{\sqrt{3}-1}\times \frac{\sqrt{3}+1}{\sqrt{3}+1}$

$=\frac{3\sqrt{3}(\sqrt{3}+1)}{2}$

$=14.14m$ (approx)

Therefore height of the balloon form the ground $=14.14+1.5=15.5m$ (approx)