A rectangular field is bounded by a fence on three sides and by a straight stream on the fourth side.

Question

A rectangular field is bounded by a fence on three sides and by a straight stream on the fourth side. Find the dimensions of the field with the maximum area that can be enclosed with 400 feet of fence. write your answer as a decimal rounded to two decimal places

smaller side = ? feet

longer side = ? feet

Answer 1

This is an optimization question from Calculus I and is solved as per the following:

Let L be the length of the rectangular area, and w the width of the area, the perimeter is therefore

L + 2w= 400 ft ( note that only one L is used since the other L is the side of the stream) so lets solve for L:

L = 400 - 2w ....eq 1 , where L is the constraint. now we need to maximize the area A = w L ...eq 2 for a rectangular area

Substitute eq 1 in 2, A(w)= w(400-2w) = 400w-2w² , we now find the derivative of the function

dA/dw = A'(w) = 400 - 4w setting this value equal to zero means that the slope of the function at point w is zero and will result in a critical point of the function. To test if this critical point is a maximum, the function must be decreasing on both sides of the point, Taking the 2nd derivative

A''(w) = -4, the value is always -ve and the function is decreasing on both sides of the critical point and the critical point is indeed a maximum

solving for w, 400 - 4w = 0, -4w = -400 , w = 100 ft, with w =100, we solve for L in eq...1 , L + 2(100) = 400, L = 400-200 = 200 ft, and the Maximum area that can be generated is

A= (200)(100) = 20000 ft²