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: \boxed{} m width: \boxed{} m

Explanation:

Step1: Define variables and fence constraint

Let $L$ = length (parallel to building), $W$ = width. Fence covers 2 widths + 1 length:
$L + 2W = 112$
Rearrange to isolate $L$: $L = 112 - 2W$

Step2: Set up area function

Area $A$ of rectangle is length × width:
$A(W) = L \times W = (112 - 2W)W = 112W - 2W^2$

Step3: Find maximum of quadratic function

The quadratic $A(W) = -2W^2 + 112W$ opens downward (coefficient of $W^2$ is negative). The vertex (maximum) occurs at $W = -\frac{b}{2a}$, where $a=-2$, $b=112$:
$W = -\frac{112}{2(-2)} = \frac{112}{4} = 28$

Step4: Calculate length

Substitute $W=28$ into $L = 112 - 2W$:
$L = 112 - 2(28) = 112 - 56 = 56$

Answer:

Length: 56 m
Width: 28 m