Solving for the Square Park Perimeter Using Path Area

This problem involves finding the perimeter of a square park given the area of a path surrounding it and the path's uniform width. Let's break down the steps to find the solution.

Understanding the Geometry

We have a square park inside a larger square that includes the path.

• Let the side length of the inner square park be '$s$' metres.
• The path has a uniform width of 2 metres.
• This means the side length of the larger square (park + path) is the side length of the park plus the width of the path on both sides.
• Side length of the outer square = $s + 2 \times 2 = s + 4$ metres.

Formulating the Area Equation

The area of the path is the difference between the area of the outer square (park + path) and the area of the inner square park.

• Area of the inner park = $s^2$ sq. metres.
• Area of the outer square = $(s + 4)^2$ sq. metres.
• Area of the path = Area of outer square - Area of inner park
• We are given that the Area of the path is 288 sq. metres.
• So, the equation is: $288 = (s + 4)^2 - s^2$.

Calculating the Side Length of the Park

Now, we solve the equation for '$s$':

1. Expand the term $(s + 4)^2$: $(s + 4)^2 = s^2 + 2(s)(4) + 4^2 = s^2 + 8s + 16$.
2. Substitute this back into the equation: $288 = (s^2 + 8s + 16) - s^2$.
3. Simplify the equation: $288 = s^2 + 8s + 16 - s^2$.
4. The '$s^2$' terms cancel out: $288 = 8s + 16$.
5. Subtract 16 from both sides: $288 - 16 = 8s$.
6. $272 = 8s$.
7. Divide by 8 to find '$s$': $s = \frac{272}{8}$.
8. $s = 34$ metres.

So, the side length of the square park is 34 metres.

Finding the Perimeter of the Park

The perimeter of a square is calculated by multiplying the side length by 4.

• Perimeter = $4 \times s$
• Perimeter = $4 \times 34$
• Perimeter = $136$ metres.

Therefore, the perimeter of the park is 136 metres.