Question

An observer on the top of a tree finds the angle of depression of a car moving towards the tree to be 30°. After 3 minutes this angle becomes 60°. After how much more time will the car reach the tree?

• A. 4 min
• B. 4.5 min
• C. 1.5 min
• D. 2 min

Hint:

When speed is constant, time is proportional to distance.

Answer 1

The Correct Option is C

Step 1: Let the height of the tree be \(h\).
\[ \tan 30^\circ = \frac{h}{x_1}, \quad \tan 60^\circ = \frac{h}{x_2} \]
Step 2:
\[ x_1 = \sqrt{3} h, \quad x_2 = \frac{h}{\sqrt{3}} \]
Step 3: Distance covered in 3 min:
\[ \sqrt{3} h - \frac{h}{\sqrt{3}} = \frac{2h}{\sqrt{3}} \]
Step 4: Remaining distance:
\[ \frac{h}{\sqrt{3}} \]
Time required: \(\frac{1}{2} \times 3 = 1.5 \text{ min}\).