Two men are on the opposite sides of a tower. They measure the angles of elevation of the top of the tower as `45^(@)` and `30^(@)` respectively. If the height of the tower is 40 m, then the distance between the men is

Options

A. 40 m
B. `40 sqrt(3)m`
C. 68.28 m
D. 109.28 m

Answer

To solve the problem, we need to find the distance between the two men standing on opposite sides of a tower, given the height of the tower and the angles of elevation they measure.

### Step-by-Step Solution:

1. **Understand the Setup**:
- Let the height of the tower be \( h = 40 \) m.
- Let the distance from the first man (who measures an angle of elevation of \( 45^\circ \)) to the base of the tower be \( a \).
- Let the distance from the second man (who measures an angle of elevation of \( 30^\circ \)) to the base of the tower be \( b \).
- The total distance between the two men is \( x = a + b \).

2. **Calculate Distance \( a \)**:
- For the first man, using the tangent of the angle of elevation:
\[ \tan(45^\circ) = \frac{h}{a} \]
- Since \( \tan(45^\circ) = 1 \):
\[ 1 = \frac{40}{a} \implies a = 40 \text{ m} \]

3. **Calculate Distance \( b \)**:
- For the second man, using the tangent of the angle of elevation:
\[ \tan(30^\circ) = \frac{h}{b} \]
- Since \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \):
\[ \frac{1}{\sqrt{3}} = \frac{40}{b} \implies b = 40\sqrt{3} \text{ m} \]

4. **Find the Total Distance \( x \)**:
- Now we can find the total distance between the two men:
\[ x = a + b = 40 + 40\sqrt{3} \]
- Factoring out the 40:
\[ x = 40(1 + \sqrt{3}) \]

5. **Approximate the Value**:
- Using \( \sqrt{3} \approx 1.732 \):
\[ 1 + \sqrt{3} \approx 1 + 1.732 = 2.732 \]
- Therefore:
\[ x \approx 40 \times 2.732 = 109.28 \text{ m} \]

### Final Answer:
The distance between the two men is approximately **109.28 meters**.