Mathematics - Application of Derivative Question with Solution | TestHub

et $h$ be the height of tree PQ and breadth of river PS be $x\mathrm{ft}$. Angle of elevation subtended by a tree is $60^{\circ }$. Also, when he retreats 20 feet, the angle becomes $30^{\circ }$.
Also, in $△\mathrm{PQS}$, $\begin{array}{rl}& \tan 60^{\circ }=\frac{h}{x}\\ & \Rightarrow h=\sqrt{3}x\end{array}$ and in $△\mathrm{PQR}$, $\begin{array}{rl}& \tan 30^{\circ }=\frac{h}{x+20}\Rightarrow \frac{1}{\sqrt{3}}=\frac{h}{x+20}\\ & \Rightarrow x+20=\sqrt{3}h\\ & \Rightarrow x+20=3x\Rightarrow 2x=20\Rightarrow x=10\end{array}$ Hence breadth of river is $10\mathrm{ft}$.