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 perimeter

Let the length of the playground be \( l \) (parallel to the building) and the width be \( w \) (perpendicular to the building). The total fence used is \( l + 2w=112 \), so \( l = 112 - 2w \).

Step2: Area formula

The area \( A \) of the rectangle is \( A=l\times w=(112 - 2w)w = 112w-2w^{2} \).

Step3: Maximize the quadratic function

The quadratic function \( A(w)=- 2w^{2}+112w \) has \( a=-2 \), \( b = 112 \). The vertex of a quadratic \( ax^{2}+bx + c \) is at \( w=-\frac{b}{2a} \). So \( w=-\frac{112}{2\times(-2)}=\frac{112}{4} = 28 \).

Step4: Find length

Substitute \( w = 28 \) into \( l=112 - 2w \), we get \( l=112-2\times28=112 - 56 = 56 \).

Answer:

Length: \( \boldsymbol{56} \) m
Width: \( \boldsymbol{28} \) m