A man standing at the top of a tower finds that a boat coming towards the tower takes 10 minutes for the angle of depression to change from 30° to 60°. How much more time will be taken by the boat to reach the tower?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 5 minutes
The angle of depression of the boat from the top T equals the angle of elevation of T from the boat (alternate angles). So the boat at P (30°) and at Q (60°) gives two right triangles on the tower TB of height \(h\), and each distance from the foot B is found with \(\tan\).
At Q: \(\displaystyle \tan60^\circ=\frac{h}{BQ}\), so \(\displaystyle BQ=\frac{h}{\sqrt3}\)
At P: \(\displaystyle \tan30^\circ=\frac{h}{BP}\), so \(BP=h\sqrt3\)
Distance covered in 10 minutes: \(\displaystyle PQ=h\sqrt3-\frac{h}{\sqrt3}=\frac{3h-h}{\sqrt3}=\frac{2h}{\sqrt3}\)
Distance still to go, \(\displaystyle BQ=\frac{h}{\sqrt3}\), is half of PQ
At the same speed, half the distance takes half the time: \(\frac{10}{2}=5\) minutes
Tip: Whenever the angle changes from 30° to 60°, the distance left is half the distance already covered, whatever the height of the tower; the remaining time is half the time taken.