The angle of elevation of the top of an incomplete vertical pillar at a horizontal distance of 100 m from its base is `45^(@)`. If the angle of elevation of the top of the complete pillar at the same point is to be `60^(@)`, then the height of the incomplete pillar is to be increased by

Options

A. `50 sqrt(2)`m
B. 100 m
C. `100(sqrt(3)-1)`m
D. `100(sqrt(3)+1)` m

Answer

To solve the problem step by step, we will use the information given about the angles of elevation and the horizontal distance from the base of the pillar.

### Step-by-Step Solution:

1. **Identify the Given Information**:
- Horizontal distance from the base of the pillar (d) = 100 m
- Angle of elevation of the incomplete pillar (θ1) = 45°
- Angle of elevation of the complete pillar (θ2) = 60°

2. **Calculate the Height of the Incomplete Pillar**:
- For the incomplete pillar, we can use the tangent of the angle of elevation:
\[ \tan(θ_1) = \frac{h_1}{d} \]
- Substituting the known values:
\[ \tan(45°) = \frac{h_1}{100} \]
- Since \(\tan(45°) = 1\):
\[ 1 = \frac{h_1}{100} \]
- Therefore, we find:
\[ h_1 = 100 \text{ m} \]

3. **Calculate the Height of the Complete Pillar**:
- For the complete pillar, we again use the tangent of the angle of elevation:
\[ \tan(θ_2) = \frac{h_2}{d} \]
- Substituting the known values:
\[ \tan(60°) = \frac{h_2}{100} \]
- Since \(\tan(60°) = \sqrt{3}\):
\[ \sqrt{3} = \frac{h_2}{100} \]
- Therefore, we find:
\[ h_2 = 100\sqrt{3} \text{ m} \]

4. **Calculate the Increase in Height**:
- The increase in height (Δh) needed to complete the pillar is given by:
\[ Δh = h_2 - h_1 \]
- Substituting the values we found:
\[ Δh = 100\sqrt{3} - 100 \]
- Factor out the common term:
\[ Δh = 100(\sqrt{3} - 1) \]

5. **Conclusion**:
- The height of the incomplete pillar needs to be increased by \(100(\sqrt{3} - 1)\) meters.