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Diana has available 200 yards of fencing and wishes to enclose a rectangular area. (a) Express the area A of the rectangle as a function of the width W of the rectangle. (b) For what value of W is the area largest? (c) What is the maximum area? (a) Express the area as a function of the width. A(w) =[] (b) For what value of W is the area largest? W- yards (Simplifty your answer.) (c) What is the maximum area?

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Final answer:

The maximum area Diana can enclose with 200 yards of fencing is found by expressing the area as a function of the width, determining the width that produces the largest area using the vertex formula for a parabola, and then calculating the area with that width, which results in a maximum area of 2500 square yards when the width is 50 yards.

Step-by-step explanation:

Diana has 200 yards of fencing to enclose a rectangular area. The perimeter (P) of a rectangle is given by P = 2l + 2w, where l is the length and w is the width. Since Diana has 200 yards of fencing, P = 200, which means 200 = 2l + 2w. To express the area A as a function of the width W, we first solve the perimeter equation for l: l = 100 - w. The area of a rectangle is A = lw, so we substitute for l to obtain A(w) = w(100 - w).

To find the value of W where the area is the largest, we need to find the vertex of the parabola A(w) = w(100 - w). This is a quadratic function in the form of A(w) = -w^2 + 100w. The vertex of a parabola given by A(w) = aw^2 + bw + c occurs at w = -b/2a. Here, a = -1 and b = 100, which gives us W = -100/(-2*1) = 50 yards.

The maximum area occurs when W = 50 yards. Substituting this back into the area function gives A(50) = 50(100 - 50) = 50 * 50 = 2500 square yards, which is the maximum area Diana can enclose with 200 yards of fencing.

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