Q.7 A footpath of uniform width runs round the inside of a rectangular field 60 m long and 40 m wide. If the path occupies 352m², find the width of the footpath. 656 cm².

To solve the problem of finding the width of the footpath, we need to work through the geometry of the situation.

### Step 1: Define the problem
- The area of the rectangular field is 60 m by 40 m.
- Let the width of the footpath be \( x \) meters.

### Step 2: Calculate the area of the field without the path
If the footpath surrounds the inside of the field, the dimensions of the area inside the footpath will be reduced by \( 2x \) in both length and width (since the path runs along all sides). 

- Length of the inner rectangle = \( 60 - 2x \) meters.
- Width of the inner rectangle = \( 40 - 2x \) meters.

So, the area of the inner rectangle is:
\[
\text{Inner Area} = (60 - 2x) \times (40 - 2x)
\]

### Step 3: Calculate the area of the path
The area of the original field is:
\[
\text{Total Area} = 60 \times 40 = 2400 \text{ m}^2
\]

The area occupied by the path is given as 352 m².

So, the area of the inner rectangle plus the area of the path is:
\[
\text{Area of the path} = \text{Total Area} - \text{Inner Area} = 2400 - (60 - 2x)(40 - 2x)
\]

We know this is equal to 352 m², so:
\[
2400 - (60 - 2x)(40 - 2x) = 352
\]

### Step 4: Solve the equation
Simplify and solve for \( x \):
\[
2400 - (2400 - 120x - 80x + 4x^2) = 352
\]
\[
2400 - 2400 + 200x - 4x^2 = 352
\]
\[
4x^2 - 200x + 352 = 0
\]
Divide the entire equation by 4 to simplify:
\[
x^2 - 50x + 88 = 0
\]

### Step 5: Solve the quadratic equation
To solve \( x^2 - 50x + 88 = 0 \), we can use the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \( a = 1 \), \( b = -50 \), and \( c = 88 \).

\[
x = \frac{50 \pm \sqrt{2500 - 352}}{2}
\]
\[
x = \frac{50 \pm \sqrt{2148}}{2}
\]
\[
x = \frac{50 \pm 46.34}{2}
\]

This gives two solutions:
\[
x = \frac{50 + 46.34}{2} = 48.17 \text{ m (not possible, as it exceeds dimensions)}
\]
\[
x = \frac{50 - 46.34}{2} = 1.83 \text{ m}
\]

### Final Answer:
The width of the footpath is approximately **1.83 meters**.

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Do you need further details or have any questions?

### Related Questions:
1. How would the solution change if the dimensions of the rectangular field were different?
2. What if the area of the path was given as 500 m² instead of 352 m²?
3. How would you solve the problem if the path was outside the rectangular field instead of inside?
4. What if the footpath was not of uniform width, how would you approach the problem?
5. Can you derive a general formula for the width of the footpath given any rectangular field and any area?

**Tip:** When dealing with quadratic equations, always consider both solutions and verify which is applicable in the given context.

Learn how to calculate the width of a footpath around a rectangular field using geometry and quadratic equations. This problem involves finding the area of the field with and without the footpath, solving a quadratic equation, and applying geometric principles. Suitable for high school students.

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