a preschool is building a new fenced playground along one side of the building. they have 112 meters of…
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
Answer
Explanation:
Step1: Set up the equations
Let the length of the playground be $l$ and the width be $w$. Since the fence is used for three - sides and the total length of the fence is 112 meters, we have $l + 2w=112$, so $l = 112 - 2w$. The area of the rectangle $A=l\times w=(112 - 2w)w=112w-2w^{2}$.
Step2: Find the derivative of the area function
The derivative of $A(w)=112w - 2w^{2}$ with respect to $w$ is $A'(w)=\frac{d}{dw}(112w-2w^{2})=112 - 4w$.
Step3: Set the derivative equal to zero
To find the critical points, we set $A'(w) = 0$. So $112 - 4w=0$. Solving for $w$ gives $4w=112$, and $w = 28$.
Step4: Find the length
Substitute $w = 28$ into the equation $l=112 - 2w$. Then $l=112-2\times28=112 - 56 = 56$.
Answer:
Length: 56 m Width: 28 m