Question

An aeroplane at an altitude of $250\text{m}$ observes the angle of depression of two boats on the opposite banks of a river to be ${45}^{\circ }$ and ${60}^{\circ }$ respectively. Find the width of the river (in $\text{m}$). Write the answer correct to the nearest whole number.

Answer 1

Let $AD=250\text{m}$, height of aeroplane
Two boats are at $B$ and $C$.
Let $BD=x$ and $DC=y$
From $△ADB,\frac{x}{250}=\cot {45}^{\circ }$
$\frac{x}{250}=1$
$\Rightarrow x=250\text{m}$
From $△ADC,\frac{y}{250}=\cot {60}^{\circ }$
$\frac{y}{250}=\frac{1}{\sqrt{3}}$
$\Rightarrow y=250\times \frac{1}{\sqrt{3}}$
Width of river $BC=BD+DC=x+y$
$=250+\frac{250}{\sqrt{3}}$
$=250(1+\frac{1}{\sqrt{3}})=250\left(\frac{\sqrt{3}+1}{\sqrt{3}}\right)$
$=250\left(\frac{1.732+1}{1.732}\right)=250\left(\frac{2.732}{1.732}\right)$
$=250\times 1.577$
$=394.25\text{m}\approx 394\text{m}$.