A man standing on the bank of a river observes that the angle subtended by a tree standing on the opposite bank is 60° on his side of bank. When he moved away 24 m from the bank, he finds the angle to be 30°. Find the breadth of the river:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 12
Let the tree QT have height \(h\) and the breadth of the river be \(b\) m. From each point where the man stands, the tree and the ground form a right triangle, so \(\tan(\text{angle})=\frac{\text{height}}{\text{distance from the tree}}\).
From the bank (P), angle 60°:
\(\tan60^\circ=\displaystyle\frac{h}{b}\), so \(h=\sqrt{3}\,b\) (1)
From 24 m back (R), angle 30°:
\(\tan30^\circ=\displaystyle\frac{h}{b+24}\), so \(h=\displaystyle\frac{b+24}{\sqrt{3}}\) (2)
Equating (1) and (2), \(h\) cancels:
\(\sqrt{3}\,b=\displaystyle\frac{b+24}{\sqrt{3}}\)
\(3b=b+24\)
\(b=12\)
The river is 12 m wide (and the tree is \(12\sqrt{3}\approx20.8\) m tall).