Question

a square playground is surrounded by a sidewalk. the sidewalk is 2n + 3 yd long on each side of the playground. the sidewalk is 0.5 yd wide. write two equivalent expressions for the perimeter of the playground. show your work.

Explanation:

Step1: Determine the side length of the playground

The sidewalk is 0.5 yd wide on each side, so the side length of the playground is the side length of the sidewalk minus twice the width of the sidewalk (once for each side). The side length of the sidewalk is \(2n + 3\) yd, so the side length of the playground \(s=(2n + 3)-2\times0.5\).
Simplify: \(s = 2n+3 - 1=2n + 2\) yd.

Step2: Find the perimeter of the playground (First expression)

The perimeter \(P\) of a square is given by \(P = 4\times\) side length. Using the side length \(s = 2n+2\), we get \(P = 4(2n + 2)\).
Expand the expression: \(P=8n + 8\) yd.

Step3: Find the second equivalent expression

Factor out 4 from \(8n + 8\): \(P = 4\times2(n + 1)=8(n + 1)\) yd. (Or we can also start from the perimeter formula with the simplified side length in a different factored form, but the key is to show two equivalent expressions. Another way: from \(P = 4(2n + 2)\), we can also distribute as we did, and factoring is another equivalent form.)

Answer:

Two equivalent expressions for the perimeter of the playground are \(4(2n + 2)\) yards and \(8n + 8\) yards (or \(8(n + 1)\) yards as another equivalent form).