Question

sienna has 1600 yards of fencing to enclose a rectangular area. find the dimensions of the rectangle that maximize the enclosed area. what is the maximum area? a rectangle that maximizes the enclosed area has a length of □ yards and a width of □ yards. the maximum area is □ square yards

Explanation:

Step1: Set up the equations

Let the length of the rectangle be \(x\) yards and the width be \(y\) yards. The perimeter formula is \(P = 2x+2y\). Given \(P=1600\), so \(2x + 2y=1600\), which simplifies to \(y = 800 - x\). The area formula is \(A=xy\). Substitute \(y = 800 - x\) into the area formula: \(A(x)=x(800 - x)=800x - x^{2}\).

Step2: Find the vertex of the quadratic function

The quadratic function \(A(x)=-x^{2}+800x\) is in the form \(y = ax^{2}+bx + c\) with \(a=-1\), \(b = 800\). The \(x\) - coordinate of the vertex of a quadratic function \(y=ax^{2}+bx + c\) is given by \(x=-\frac{b}{2a}\).
Substitute \(a=-1\) and \(b = 800\) into \(x=-\frac{b}{2a}\):
\(x=-\frac{800}{2\times(-1)}=\frac{- 800}{-2}=400\).
Since \(y = 800 - x\), when \(x = 400\), \(y=800 - 400=400\).

Step3: Calculate the maximum area

Substitute \(x = 400\) and \(y = 400\) into the area formula \(A=xy\).
\(A=(400)\times(400)=160000\).

Answer:

A rectangle that maximizes the enclosed area has a length of \(400\) yards and a width of \(400\) yards. The maximum area is \(160000\) square yards.