A ladder makes an angle of `30^(@)` with a wall. If the foot of the ladder is 5 m away from the wall, find the length of the ladder.

Answer

To solve the problem step by step, we will use trigonometric properties in a right triangle formed by the ladder, the wall, and the ground.

### Step-by-Step Solution:

1. **Draw a Diagram**:
- Draw a vertical line representing the wall.
- Draw a horizontal line representing the ground.
- Draw the ladder leaning against the wall, forming a right triangle with the wall and the ground.

2. **Identify the Components**:
- Let point A be the point where the ladder touches the wall.
- Let point B be the foot of the ladder on the ground.
- Let point C be the point on the ground directly below point A.
- The distance from point B to the wall (point C) is given as 5 m.

3. **Identify the Angle**:
- The ladder makes an angle of \(30^\circ\) with the wall.

4. **Use Trigonometric Ratios**:
- In the right triangle ABC, we can use the sine function:
\[ \sin(30^\circ) = \frac{\text{Opposite (BC)}}{\text{Hypotenuse (AC)}} \]
- Here, BC is the height of the ladder on the wall, and AC is the length of the ladder.

5. **Calculate the Sine of 30 Degrees**:
- We know that:
\[ \sin(30^\circ) = \frac{1}{2} \]

6. **Set Up the Equation**:
- Substitute the known values into the sine equation:
\[ \frac{1}{2} = \frac{BC}{AC} \]
- We also know that the distance from the foot of the ladder to the wall (BC) is 5 m:
\[ \frac{1}{2} = \frac{5}{AC} \]

7. **Solve for AC (Length of the Ladder)**:
- Rearranging the equation gives:
\[ AC = 5 \times 2 = 10 \text{ m} \]

8. **Conclusion**:
- The length of the ladder is 10 meters.

### Final Answer:
The length of the ladder is **10 meters**.