Question

A moving boat is observed from the top of a $150m$ high cliff moving away from the cliff. The angle of depression of the boat changes from ${60}^o$ to ${45}^{}$ in $2$ minutes. Find the speed of the boat in $m/h$.

Answer 1

In $\Delta ABC$ $tan{60}^{\circ }=\frac{150}{x}$

$\sqrt{3}=\frac{150}{x}$

$x=\frac{150}{\sqrt{3}}\times \frac{\sqrt{3}}{\sqrt{3}}=50\sqrt{3}m$

$tan{45}^{\circ }=\frac{150}{120s+st}$

$120s+st=150$

$120s=150-50\sqrt{3}=50(3-\sqrt{3})$

$120s=50\left(1.268\right)=63.4$

$s=\frac{63.4}{120}=\frac{634}{1200}=\frac{317}{600}$

$s=\frac{317}{600}\times 3600=1902m/hr$