How do you solve this optimization question?

A farmer wants to fence an area of #13.5# million square feet in a rectangular field and then divide it in half with a fence parallel to one of the sides of the rectangle.

What should the lengths of the sides of the rectangular field be so as to minimize the cost of the fence?

1 Answer
Douglas K.
Apr 23, 2018

I shall prove that the minimum perimeter of a rectangle of any given area where the sides are not fixed is a square.

This equation describes the perimeter:

#P= 2L + 2W" [1]"#

This equation describes the area with a given constant:

#"Area" = LW = C" [2]"#

Substitute #W = C/L# into equation [1]:

#P= 2L + 2C/L#

Compute the derivative with respect to L:

#(dP)/(dL) = 2 - 2C/L^2#

Set the first derivative equal to 0:

#2 - 2C/L^2 = 0#

#-2C/L^2= -2#

#C/L^2= 1#

#L^2 = C#

#L = sqrtC" [3]"#

NOTE: L must be positive, therefore, the traditional #+-# in front of the radical makes no sense.

Compute the second derivative:

#(d^2P)/(dL^2) =4C/L^3#

Perform the second derivative test:

#(d^2P)/(dL^2) =4C/C^(3/2) = 4sqrtC/C#

The second derivative test tells us that equation [3] is a minimum.

Substitute equation [3] into equation [2]:

#sqrtCW = C#

#W = C/sqrtC#

#W = sqrtC#

Please observe that #L= W= sqrtC# describes the special case of a rectangle called, a square. Therefore, it does not matter whether the dividing fence is parallel to the length or the width because they are equal.

In the case were #C = 13.5 xx 10^6" ft"^2#

#L = W = sqrt(13.5 xx 10^6" ft"^2)#

#L = W ~~ 3674.23" ft"#

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