Question

The angle of elevation from a point on the bank of a river to the top of a temple on the other bank is ${45}^o$. Retreating $50m$, the observer finds the new angle of elevation as ${30}^{\circ }$. What is the width of the river ?

• A. $50m$
• B. $50\sqrt{3}m$
• C. $\frac{50}{(\sqrt{3}-1)}m$
• D. $100m$

Answer 1

Answer: (C)

$\frac{50}{(\sqrt{3}-1)}m$

Let the height of the temple be h.
And let length of the bank of the river be y.
Then
$\tan {45}^0=\frac{h}{y}=1$
Hence
$h=y$ ...(i)
and
$\tan {30}^0=\frac{1}{\sqrt{3}}=\frac{h}{y+50}$
Or
$\sqrt{3}y=y+50$ ...$(h=y)$ from $\left(i\right)$
$y=\frac{50}{\sqrt{3}-1}m$