Final answer:
The area of the rectangle Diana wants to enclose is a function of its width, with the maximum area obtained when the width is 30 yards, resulting in a maximum area of 1800 square yards.
Step-by-step explanation:
Rectangular Area Optimization
Diana has 120 yards of fencing to enclose a rectangular area. The total perimeter (P) of the rectangle is the sum of twice its width (W) and twice its length (L), so we have the equation P = 2W + 2L. Given that P is 120 yards, we can express L in terms of W: L = (120 - 2W) / 2.
(a) The area (A) of the rectangle can be expressed as A = W × L. Substituting L, we get A(W) = W × ((120 - 2W) / 2).
(b) To find the value of W that makes the area largest, we take the derivative of A with respect to W and set it to zero. This gives us W = 30 yards.
(c) The maximum area can then be found by substituting W back into the area function, yielding maximum area = 1800 square yards.