Question

A person standing on the bank of a river observes that the angle of elevation of the top of a tree standing on the opposite bank is 60^o, when he moves 40 m away from the bank, he finds the angle of elevation to be 30^o. Find the height of the tree and width of the river.

Answer 1

Let:
AB = h (Height of the tree)
Let x m be the width of the river bank.
Now,
$$\begin{gathered} \mathrm{In}∆ACB, \\ \tan 60°=\frac{AB\hspace{0.17em}}{CB}=\frac{h}{x} \\ \Rightarrow \sqrt{3}=\frac{h}{x} \\ \Rightarrow h=\sqrt{3}x...\left(\mathrm{i}\right) \\ \mathrm{In}∆ABD, \\ \tan 30°=\frac{AB}{BD}=\frac{h}{40+x} \\ \Rightarrow \frac{1}{\sqrt{3}}=\frac{h}{40+x} \\ \Rightarrow 40+x=\sqrt{3}h \\ \Rightarrow 40+x=\sqrt{3}\left(\sqrt{3}x\right)\left[\mathrm{From}\left(\mathrm{i}\right)\right] \\ \Rightarrow 40+x=3x \\ \Rightarrow 40=2x \\ \Rightarrow 20=x \\ \therefore h=\sqrt{3}\times 20=34.6 \\ \mathrm{Thus},\mathrm{the}\mathrm{height}\mathrm{of}\mathrm{the}\mathrm{tree}\mathrm{is}34.6\mathrm{m}\mathrm{and}\mathrm{the}\mathrm{width}\mathrm{of}\mathrm{the}\mathrm{river}\mathrm{bank}\mathrm{is}20\mathrm{m}. \end{gathered}$$