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 two men standing on opposite sides of a tower, given the angles of elevation to the top of the tower and the height of the tower.

### Step-by-Step Solution:

1. **Understanding the Problem:**
- Let the height of the tower be \( h = 40 \) m.
- Let the distance from the first man to the base of the tower be \( a \).
- Let the distance from the second man to the base of the tower be \( b \).
- The angle of elevation from the first man is \( 45^\circ \) and from the second man is \( 30^\circ \).

2. **Using Trigonometric Ratios:**
- For the first man (angle of elevation \( 45^\circ \)):
\[ \tan(45^\circ) = \frac{\text{height}}{\text{distance from tower}} = \frac{h}{a} \]
Since \( \tan(45^\circ) = 1 \):
\[ 1 = \frac{40}{a} \implies a = 40 \text{ m} \]

3. **Calculating the Distance for the Second Man:**
- For the second man (angle of elevation \( 30^\circ \)):
\[ \tan(30^\circ) = \frac{\text{height}}{\text{distance from tower}} = \frac{h}{b} \]
Since \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \):
\[ \frac{1}{\sqrt{3}} = \frac{40}{b} \implies b = 40\sqrt{3} \text{ m} \]

4. **Finding the Total Distance Between the Two Men:**
- The total distance \( D \) between the two men is given by:
\[ D = a + b = 40 + 40\sqrt{3} \]
- Factoring out the common term:
\[ D = 40(1 + \sqrt{3}) \]

5. **Calculating the Numerical Value:**
- Using the approximate value \( \sqrt{3} \approx 1.732 \):
\[ D \approx 40(1 + 1.732) = 40 \times 2.732 \approx 109.28 \text{ m} \]

### Final Answer:
The distance between the two men is approximately \( 109.28 \) meters.