Question

The angle of elevation of the top of a pole from a point A on the ground is 30°. The angle of elevation changes to 45°, after moving 20 metres towards the base of the pole. Then the height of the pole, in metres, is:

• A. \(30\)
• B. \(15(\sqrt{5} + 1)\)
• C. \(20(\sqrt{3} + 1)\)
• D. \(10(\sqrt{3} + 1)\)

Hint:

For problems involving angles of elevation, use the tangent function and set up equations based on the known angles to solve for the height or distance.

Answer 1

The Correct Option is D

Step 1: Let the height of the pole be \(h\) and the distance of the point A from the base be \(x\). From the given information: 1. When the angle of elevation is 30°, we have: \[ \tan 30^\circ = \frac{h}{x} \quad \Rightarrow \quad h = x \cdot \frac{1}{\sqrt{3}} \quad \cdots (1) \] 2. When the angle of elevation becomes 45° after moving 20 meters towards the base, the new distance from the pole is \(x - 20\), and we have: \[ \tan 45^\circ = \frac{h}{x - 20} \quad \Rightarrow \quad h = x - 20 \quad \cdots (2) \] Equating (1) and (2): \[ x \cdot \frac{1}{\sqrt{3}} = x - 20 \quad \Rightarrow \quad x = 20(\sqrt{3} + 1) \] Substituting this value of \(x\) in (1): \[ h = 20(\sqrt{3} + 1) \cdot \frac{1}{\sqrt{3}} = 10(\sqrt{3} + 1) \]