Question

a preschool is building a new fenced playground along one side of the building. they have 112 meters of fence to use. the fence will surround three sides of the rectangular playground. the figure shows the plan for the fence. what length and width of the fence would maximize the area of the playground? enter the answer in the boxes. length: m width: m

Explanation:

Step1: Set up the equations

Let the length be \(x\) and the width be \(y\). The total length of the fence is \(2y + x=112\), so \(x = 112 - 2y\). The area \(A=xy=y(112 - 2y)=112y-2y^{2}\).

Step2: Find the maximum of the quadratic function

For a quadratic function \(A(y)=- 2y^{2}+112y\), where \(a=-2\), \(b = 112\). The vertex of a quadratic function \(y=ax^{2}+bx + c\) occurs at \(y=-\frac{b}{2a}\).
Substitute \(a=-2\) and \(b = 112\) into \(y =-\frac{b}{2a}\), we get \(y=-\frac{112}{2\times(-2)} = 28\).

Step3: Find the value of \(x\)

Substitute \(y = 28\) into \(x=112 - 2y\). Then \(x=112-2\times28=56\).

Answer:

Length: \(56\) m
Width: \(28\) m