The angle of elevation of the top of a tower from a point 40 m away from its foot is `60^(@)`. Find the height of the tower.

Answer

To find the height of the tower based on the given information, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Triangle**:
- Let \( A \) be the top of the tower, \( B \) be the point on the ground where the angle of elevation is measured, and \( C \) be the foot of the tower.
- We know that \( BC = 40 \) m (the distance from the point to the foot of the tower) and the angle of elevation \( \angle ABC = 60^\circ \).

2. **Use the Tangent Function**:
- In triangle \( ABC \), we can use the tangent function, which relates the opposite side (height of the tower \( AC \)) to the adjacent side (distance from the point to the foot of the tower \( BC \)).
- The formula is given by:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]
- Here, \( \theta = 60^\circ \), so we have:
\[ \tan(60^\circ) = \frac{AC}{BC} \]

3. **Substitute Known Values**:
- We know that \( \tan(60^\circ) = \sqrt{3} \) and \( BC = 40 \) m. Therefore, we can write:
\[ \sqrt{3} = \frac{AC}{40} \]

4. **Solve for the Height of the Tower (AC)**:
- Rearranging the equation to find \( AC \):
\[ AC = 40 \cdot \sqrt{3} \]

5. **Calculate the Height**:
- Now, we can compute the height:
\[ AC = 40\sqrt{3} \text{ m} \]

6. **Final Answer**:
- Thus, the height of the tower is:
\[ \text{Height of the tower} = 40\sqrt{3} \text{ m} \]