The top of a 15 metre high tower makes an angle of elevation of 60° with the bottom of an electric pole and an angle of elevation of 30° with the top of the pole, then height of the electric pole is :

Question

Accepted Answer

To determine the height of the electric pole, we can use trigonometric principles involving angles of elevation and the properties of right triangles.

Given:
- The height of the tower is 15 meters.
- The angle of elevation from the bottom of the pole to the top of the tower is 60°.
- The angle of elevation from the top of the pole to the top of the tower is 30°.

Let's denote:
- $ h $ as the height of the electric pole.
- $ d $ as the horizontal distance between the tower and the pole.

### Step-by-step Explanation:

1. **Using the 60° Angle of Elevation:**
   - The angle of elevation from the bottom of the pole to the top of the tower is 60°.
   - In the right triangle formed, the opposite side is the height of the tower (15 m), and the adjacent side is the distance $ d $.
   - Using the tangent function: 
     $$
     \tan(60°) = \frac{\text{opposite}}{\text{adjacent}} = \frac{15}{d}
     $$
   - We know $\tan(60°) = \sqrt{3}$, so:
     $$
     \sqrt{3} = \frac{15}{d} \implies d = \frac{15}{\sqrt{3}} = 5\sqrt{3}
     $$

2. **Using the 30° Angle of Elevation:**
   - The angle of elevation from the top of the pole to the top of the tower is 30°.
   - In this scenario, the height difference between the tower and the pole is $ 15 - h $.
   - Using the tangent function:
     $$
     \tan(30°) = \frac{\text{opposite}}{\text{adjacent}} = \frac{15 - h}{d}
     $$
   - We know $\tan(30°) = \frac{1}{\sqrt{3}}$, so:
     $$
     \frac{1}{\sqrt{3}} = \frac{15 - h}{5\sqrt{3}}
     $$
   - Solving for $ h $:
     $$
     15 - h = \frac{5\sqrt{3}}{\sqrt{3}} = 5
     $$
     $$
     h = 15 - 5 = 10
     $$

Thus, the height of the electric pole is 10 meters, which corresponds to Option 3.

### Explanation for Incorrect Options:

- **Option 1 (8 m):** This option is incorrect because the calculations based on the given angles and the height of the tower do not result in a height of 8 meters for the pole.
- **Option 2 (9 m):** This option is incorrect for the same reason as above; the trigonometric calculations do not support a height of 9 meters.
- **Option 4 (12 m):** This option is incorrect because the derived height from the trigonometric relationships does not match 12 meters.

Therefore, the correct answer is indeed Option 3, which is 10 meters.