Question

Has available 480480 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?

Answer 1

I assume you meant 480 yards and not 480480 yards.

a) 2L + 2W = 480 so 2L = 480 - 2W; L = 240 - W; The area of the rectangle (A) is LW; A = LW; A = (240 - W)W; A = 240W - W^2; F(W) = 240W - W^2.

b) The maximum area of the rectangle is when the derivative of F(W) = 0. F'(W) = 240 - 2W ; 0 = 240 - 2W; W = 120. The area of the rectangle is greatest when W = 120.

c) The maximum area is A = (120)(120) = 14400 ft^2