From a point on a bridge across a river, the angles of depression of the banks on opposite sides of the river are 30° and 45°, respectively. If the bridge is at a height of 9 m from the surface of the river, then find the width of the river.
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(9(\sqrt{3}+1)\)
The angle of depression from the point on the bridge to a bank is equal to the angle of elevation of that point seen from the bank (alternate angles). In each right triangle, \(\tan\theta=\frac{\text{height}}{\text{distance along the river}}\), and the two distances, on opposite sides of the foot F of the point, add up to the width.
Bank A, angle 30°: \(\displaystyle \tan30^\circ=\frac{9}{AF}\), so \(AF=\frac{9}{1/\sqrt{3}}=9\sqrt{3}\,\mathrm{m}\)
Bank B, angle 45°: \(\displaystyle \tan45^\circ=\frac{9}{FB}\), so \(FB=9\,\mathrm{m}\)
Width of the river \(=AF+FB=9\sqrt{3}+9=9(\sqrt{3}+1)\,\mathrm{m}\)