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Question

A rancher has 800 feet of fencing to put around a rectangular field and then subdivide the field into 3 identical smaller rectangular plots by placing two fences parallel to one of the field's shorter sides. Find the dimensions that maximize the enclosed area. Write your answers as fractions reduced to lowest terms.

Answer 1

Let x be the dimension of the shorter side, and y be be the dimension of the longer side of the field.

 

There are 4 fences along the shorter side, and 2 fences along the longer side, so

4x + 2y = 800 ⇒ y = 400 − 2x

 

The area of the rectangular field is

A = xy

   = x(400 − 2x)

   = − 2x² + 400x

 

The area is a parabola opens downward, so the maximum area occurs at the parabola vertex.

At the vertex

x = −b/2a

   = −400/[2(−2)]

   = 100

 

y = 400 −2x

   = 200

 

So the dimensions of the rectangular field that maximize the enclosed area is 100 ft x 200 ft.

 

 

Answer 2

 

  So there are 3 widths and 2 lengths

 

 Perimeter = 3 x width + 2 * length

 

 

 800 = 3*W + 2* L --->  L = (800 - 3*w)/2 = 400 - (3/2)w

 

Area = length * width = [400 - (3/2)w]*w = A(w) <--- Area function in terms of the width and it is a parabola

 

Typically, to find the extrema without calculus, the parabola is rewritten in the form a(x-h)^2+ k 

by completing the square, so that the vertex is (h,k). Another way is to average the zeros.

 

The two zeros of this parabola are:   400 - (3/2)w = 0  and w = 0

 

                                                    400 = (3/2)w   and w = 0

 

                                                   2*400/3 = w  and w = 0

 

                                                       800/3 = w and w=0

 

The axis of symmetry is the average of the zeros: ( 0 + 800/3) divided by 2 = 800/3 divided by 2

 

                                                                                                              = 800/3 * 1/2 = 400/3

 

The max occurs at w=400/3.

 

L = 400 - (3/2)(400/3) = 400 - 200 = 200.

 

The max area is 400/3*200 = 80000/3 = 26666 and 2/3 square feet

 

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