The angle of elevation of the top of an incomplete vertical pillar at a horizontal distance of 50 m from its base is `45^@`. If the angle of elevation of the top of the complete pillar the same point is to be `60^@`,then the height of the incomplete pillar is to be increased by

Options

A. `50sqrt2`m
B. 100 m
C. `100 (sqrt3-1)m`
D. `100(sqrt3+1)`m

Answer

To solve the problem step by step, we will use the concepts of trigonometry, particularly the tangent function, to find the heights of the incomplete and complete pillars.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We have an incomplete vertical pillar and a complete vertical pillar.
- The horizontal distance from the observer to the base of the pillar is 50 m.
- The angle of elevation to the top of the incomplete pillar is \(45^\circ\).
- The angle of elevation to the top of the complete pillar is \(60^\circ\).

2. **Finding the Height of the Incomplete Pillar**:
- Let the height of the incomplete pillar be \(h\).
- Using the tangent of the angle of elevation:
\[ \tan(45^\circ) = \frac{h}{50} \]
- We know that \(\tan(45^\circ) = 1\), so:
\[ 1 = \frac{h}{50} \]
- Thus, solving for \(h\):
\[ h = 50 \text{ m} \]

3. **Finding the Height of the Complete Pillar**:
- Let the height of the complete pillar be \(H\).
- Using the tangent of the angle of elevation:
\[ \tan(60^\circ) = \frac{H}{50} \]
- We know that \(\tan(60^\circ) = \sqrt{3}\), so:
\[ \sqrt{3} = \frac{H}{50} \]
- Thus, solving for \(H\):
\[ H = 50\sqrt{3} \text{ m} \]

4. **Calculating the Increase in Height**:
- The increase in height required to complete the pillar is given by:
\[ \Delta H = H - h \]
- Substituting the values we found:
\[ \Delta H = 50\sqrt{3} - 50 \]
- Factoring out 50:
\[ \Delta H = 50(\sqrt{3} - 1) \]

5. **Final Result**:
- The height of the incomplete pillar needs to be increased by:
\[ \Delta H = 50(\sqrt{3} - 1) \text{ m} \]