A footpath of uniform width runs all around the inside of a rectangular field 38 m long and 32 m wide. If the path occupies 600 `m^(2)`, find its width.

Answer

To solve the problem step by step, we need to find the width of the footpath that runs around the inside of a rectangular field. Here’s how to approach it:

### Step 1: Understand the dimensions of the field
The rectangular field has a length of 38 meters and a width of 32 meters.

### Step 2: Define the width of the footpath
Let the width of the footpath be \( x \) meters. Since the footpath runs uniformly around the inside of the field, it reduces both the length and the width of the inner rectangle.

### Step 3: Calculate the dimensions of the inner rectangle
The dimensions of the inner rectangle (the area not occupied by the footpath) will be:
- Length: \( 38 - 2x \) (subtracting \( x \) from both ends)
- Width: \( 32 - 2x \) (subtracting \( x \) from both ends)

### Step 4: Calculate the area of the outer rectangle
The area of the outer rectangle (the entire field) is given by:
\[ \text{Area}_{\text{outer}} = \text{Length} \times \text{Width} = 38 \times 32 = 1216 \, m^2 \]

### Step 5: Calculate the area of the inner rectangle
The area of the inner rectangle is:
\[ \text{Area}_{\text{inner}} = (38 - 2x)(32 - 2x) \]

### Step 6: Set up the equation for the area of the footpath
The area of the footpath is the difference between the area of the outer rectangle and the area of the inner rectangle:
\[ \text{Area}_{\text{footpath}} = \text{Area}_{\text{outer}} - \text{Area}_{\text{inner}} = 1216 - (38 - 2x)(32 - 2x) \]
Given that the area of the footpath is 600 m², we can set up the equation:
\[ 1216 - (38 - 2x)(32 - 2x) = 600 \]

### Step 7: Simplify the equation
First, calculate the area of the inner rectangle:
\[ (38 - 2x)(32 - 2x) = 1216 - 76x + 4x^2 \]
Now substitute this back into the equation:
\[ 1216 - (1216 - 76x + 4x^2) = 600 \]
This simplifies to:
\[ 76x - 4x^2 = 600 \]
Rearranging gives:
\[ 4x^2 - 76x + 600 = 0 \]

### Step 8: Simplify the equation further
Divide the entire equation by 4:
\[ x^2 - 19x + 150 = 0 \]

### Step 9: Factor the quadratic equation
To factor \( x^2 - 19x + 150 = 0 \), we look for two numbers that multiply to 150 and add to -19. These numbers are -15 and -10:
\[ (x - 15)(x - 10) = 0 \]

### Step 10: Solve for \( x \)
Setting each factor to zero gives:
1. \( x - 15 = 0 \) → \( x = 15 \)
2. \( x - 10 = 0 \) → \( x = 10 \)

### Step 11: Check the validity of the solutions
We need to ensure that the values of \( x \) do not make the dimensions of the inner rectangle negative:
- For \( x = 15 \):
- Length: \( 38 - 2(15) = 8 \) (valid)
- Width: \( 32 - 2(15) = 2 \) (valid)

- For \( x = 10 \):
- Length: \( 38 - 2(10) = 18 \) (valid)
- Width: \( 32 - 2(10) = 12 \) (valid)

Both values are valid, but since the footpath must have a uniform width and cannot exceed half of the smaller dimension of the field, we choose \( x = 10 \) meters as the maximum feasible width.

### Final Answer
The width of the footpath is \( \boxed{10} \) meters.