A footpath of uniform width runs all around the outside of a rectangular field 30 m long and 24 m wide. If the path occupies an area of 360 `m^(2)`, find its width.

Answer

To find the width of the footpath that runs around a rectangular field, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the dimensions of the inner rectangle (field)**:
- Length (L) = 30 m
- Width (W) = 24 m

2. **Let the width of the footpath be \( x \)**.

3. **Determine the dimensions of the outer rectangle**:
- The length of the outer rectangle = \( L + 2x = 30 + 2x \)
- The width of the outer rectangle = \( W + 2x = 24 + 2x \)

4. **Calculate the area of the inner rectangle**:
- Area of the inner rectangle = \( L \times W = 30 \times 24 = 720 \, m^2 \)

5. **Calculate the area of the outer rectangle**:
- Area of the outer rectangle = \( (30 + 2x)(24 + 2x) \)

6. **Set up the equation for the area of the footpath**:
- Area of the footpath = Area of the outer rectangle - Area of the inner rectangle
- Given that the area of the footpath is 360 m², we can write:
\[ (30 + 2x)(24 + 2x) - 720 = 360 \]

7. **Simplify the equation**:
\[ (30 + 2x)(24 + 2x) = 1080 \]
Expanding the left side:
\[ 720 + 60x + 48x + 4x^2 = 1080 \]
Combine like terms:
\[ 4x^2 + 108x + 720 - 1080 = 0 \]
This simplifies to:
\[ 4x^2 + 108x - 360 = 0 \]

8. **Divide the entire equation by 4**:
\[ x^2 + 27x - 90 = 0 \]

9. **Factor the quadratic equation**:
- We need two numbers that multiply to -90 and add to 27. These numbers are 30 and -3.
\[ (x + 30)(x - 3) = 0 \]

10. **Solve for \( x \)**:
- Set each factor to zero:
\[ x + 30 = 0 \quad \Rightarrow \quad x = -30 \quad \text{(not valid, as width cannot be negative)} \]
\[ x - 3 = 0 \quad \Rightarrow \quad x = 3 \]

11. **Conclusion**:
- The width of the footpath is \( 3 \, m \).