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

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

Answer

Explanation:

Step1: Define Variables

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 we can express ( l = 112 - 2w ).

Step2: Area Function

The area ( A ) of the rectangle is ( A = l \times w ). Substitute ( l ) from Step 1: ( A(w) = (112 - 2w)w = 112w - 2w^2 ).

Step3: Maximize the Quadratic

The area function is a quadratic ( A(w) = -2w^2 + 112w ), which opens downward (since the coefficient of ( w^2 ) is negative). The vertex of a quadratic ( ax^2 + bx + c ) occurs at ( w = -\frac{b}{2a} ). Here, ( a = -2 ), ( b = 112 ), so ( w = -\frac{112}{2(-2)} = \frac{112}{4} = 28 ).

Step4: Find Length

Substitute ( w = 28 ) into ( l = 112 - 2w ): ( l = 112 - 2(28) = 112 - 56 = 56 ).

Answer:

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