sienna has 1600 yards of fencing to enclose a rectangular area. find the dimensions of the rectangle that…

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.