A ladder 15m long makes an angle of `60^(@)` with the wall. Find the height of the point , where the ladder touches the wall.

Options

A. `15sqrt(3)`
B. `(15sqrt(3))/(2) m`
C. `15/2m`
D. `15 m`

Explanation

To solve the problem of finding the height at which the ladder touches the wall, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem Setup**:
- We have a ladder (AC) that is 15 meters long.
- The ladder makes an angle of 60 degrees with the wall (AB).
- We need to find the height (h) at which the ladder touches the wall.

2. **Identify the Triangle**:
- The ladder, the wall, and the ground form a right triangle (triangle ABC) where:
- AC is the ladder (hypotenuse).
- AB is the wall (height).
- BC is the base (distance from the wall to the bottom of the ladder).

3. **Determine the Angles**:
- The angle between the ladder and the wall is 60 degrees.
- Therefore, the angle at point C (the angle opposite to the height AB) can be calculated as:
\[ \text{Angle C} = 90^\circ - 60^\circ = 30^\circ \]

4. **Use the Sine Function**:
- In triangle ABC, we can use the sine function to find the height (h):
\[ \sin(\text{Angle C}) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{h}{AC} \]
- Here, AC is the length of the ladder which is 15 meters, and the opposite side is the height (h).
- Thus, we have:
\[ \sin(30^\circ) = \frac{h}{15} \]

5. **Substitute the Value of Sine**:
- We know that:
\[ \sin(30^\circ) = \frac{1}{2} \]
- Substituting this into the equation gives:
\[ \frac{1}{2} = \frac{h}{15} \]

6. **Solve for h**:
- To find h, we can rearrange the equation:
\[ h = 15 \times \frac{1}{2} = \frac{15}{2} = 7.5 \text{ meters} \]

7. **Conclusion**:
- The height at which the ladder touches the wall is \( 7.5 \) meters.

### Final Answer:
The height of the point where the ladder touches the wall is \( 7.5 \) meters.