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 trigonometric ratios related to the angles of elevation.

### Step 1: Understand the Problem
We have a vertical pillar that is incomplete, and we need to find out how much we need to increase its height to make it complete. The angles of elevation from a point 100 meters away from the base of the pillar are given as 45 degrees for the incomplete pillar and 60 degrees for the complete pillar.

### Step 2: Set Up the Diagram
Let's denote:
- Point A: the point on the ground where the observer is standing.
- Point B: the top of the incomplete pillar.
- Point C: the top of the complete pillar.
- The height of the incomplete pillar (AB) as \( h_1 \).
- The height of the complete pillar (AC) as \( h_2 \).
- The horizontal distance from point A to the base of the pillar (point D) is 100 meters.

### Step 3: Use Trigonometry for the Incomplete Pillar
For the incomplete pillar, we have:
\[ \tan(45^\circ) = \frac{h_1}{100} \]
Since \( \tan(45^\circ) = 1 \):
\[ 1 = \frac{h_1}{100} \]
Thus, we can find \( h_1 \):
\[ h_1 = 100 \text{ meters} \]

### Step 4: Use Trigonometry for the Complete Pillar
For the complete pillar, we have:
\[ \tan(60^\circ) = \frac{h_2}{100} \]
Since \( \tan(60^\circ) = \sqrt{3} \):
\[ \sqrt{3} = \frac{h_2}{100} \]
Thus, we can find \( h_2 \):
\[ h_2 = 100\sqrt{3} \text{ meters} \]

### Step 5: Find the Increase in Height
To find how much the height of the incomplete pillar needs to be increased, we calculate:
\[ \text{Increase in height} = h_2 - h_1 \]
Substituting the values we found:
\[ \text{Increase in height} = 100\sqrt{3} - 100 \]
Factoring out 100:
\[ \text{Increase in height} = 100(\sqrt{3} - 1) \text{ meters} \]

### Final Answer
The height of the incomplete pillar needs to be increased by \( 100(\sqrt{3} - 1) \) meters.