A farmer has 360 feet of fence to enclose a rectangular area. What dimensions for the rectangle result in the maximum area enclosed by the​ fence?

Question

what are the length and width

Answer 1

A = LW

 

2(L + W) = 360 ⇒

 

L = 180 - W

 

A = (180 - W)W = -W² + 180W

 

Maximum A is at the vertex of the A-vs.-W parabola, which is at

 

W = -180/[2(-1)] = 90

 

W = 90

 

L = 180 - 90 = 90

 

Answer 2

Lw = Area

 

2L + 2w = 360

L+w=180

L = 180 - w

 

(180-w)w = Area = A(w) is a function of width w

 

 The zeros of this function is w=0 and w=180.

 The max occurs at the average of the zeros, which is 90.

 

L = 180 - 90 = 90

w = 90

 

Max area occurs with dimensions L=w=90, which is really a square.

Max area is 8100