diana has 440 yards of fencing. she wishes to enclose a rectangular area. express the area a of the…

Explanation:

Step1: Find the length in terms of width

Let the width of the rectangle be (W). The perimeter of a rectangle is (P = 2L+2W). Given (P = 440), we have (440=2L + 2W). Solving for (L), we get (L=\frac{440 - 2W}{2}=220 - W).

Step2: Express the area as a function of width

The area of a rectangle (A = L\times W). Substituting (L = 220 - W) into the area formula, we get (A(W)=(220 - W)W=220W - W^{2}).

Step3: Find the value of (W) for maximum area

For a quadratic function (y = ax^{2}+bx + c) ((a=- 1), (b = 220), (c = 0) in (A(W)=-W^{2}+220W)), the vertex of the parabola (x=-\frac{b}{2a}). Substituting (a=-1) and (b = 220) into (W=-\frac{b}{2a}), we have (W=-\frac{220}{2\times(-1)} = 110).

Step4: Find the maximum area

Substitute (W = 110) into the area formula (A(W)=220W - W^{2}). (A(110)=220\times110-110^{2}=110\times(220 - 110)=110\times110 = 12100)

Answer:

The area function is (A(W)=220W - W^{2}). The value of (W) for which the area is largest is (W = 110) yards. The maximum area is (12100) square - yards.