The village playground is shaped as a square of side 40 metres. A path of width metres is created around the playground for people to walk. Find an expression for the area of the path in terms of .

Exploring Algebraic Identities | EOT

Question 7

The village playground is shaped as a square of side 40 metres. A path of width $s$ metres is created around the playground for people to walk. Find an expression for the area of the path in terms of $s$ .

Solution

Understand the Question

The playground is an inner square with side length $40\text{ m}$ .
A path of uniform width $s\text{ metres}$ surrounds the playground on all four sides, forming a larger outer square.
The side length of the outer square increases by $s$ on both opposite sides, giving a total side length of $(40 + 2s)\text{ m}$ .
The area of the path is found by subtracting the area of the inner playground from the area of the outer square: $\text{Area of path} = \text{Area of outer square} - \text{Area of inner square}$

Step 1 · Find the Area of the Playground

Given side of the square playground $= 40\text{ m}$ .

\begin{aligned} \text{Area of playground} &= \text{side}^2 \\ &= 40^2 \\ &= 1600\text{ m}^2 \end{aligned}

Step 2 · Find the Side Length and Area of the Outer Square

Since the path of width $s\text{ m}$ extends on both sides of the square: $\text{Side of outer square} = 40 + s + s = (40 + 2s)\text{ m}$

\begin{aligned} \text{Area of outer square} &= (40 + 2s)^2 \\ &= 40^2 + 2(40)(2s) + (2s)^2 \\ &= 1600 + 160s + 4s^2 \end{aligned}

Step 3 · Calculate the Area of the Path

\begin{aligned} \text{Area of path} &= \text{Area of outer square} - \text{Area of playground} \\ &= (1600 + 160s + 4s^2) - 1600 \\ &= 4s^2 + 160s \end{aligned}

Answer

$(4s^2 + 160s)\text{ m}^2$

Common Mistakes

Adding width only once: Taking the outer square's side length as $40 + s$ instead of $40 + 2s$ . Because the border surrounds the square, width $s$ must be added to both ends of each dimension.
Squaring error: Expanding $(40 + 2s)^2$ incorrectly as $40^2 + (2s)^2 = 1600 + 4s^2$ , forgetting the middle term $2(40)(2s) = 160s$ .

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