Question

A farmer wants to fence an area of 6 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. How can he do this so as to minimize the cost of the fence?

Answer 1

If the rectangular field has notional sides `x` and `y`, then it has area:

• `A(bbx) = xy qquad [= 6*10^6 " sq ft"]`

The length of fencing required, if `x` is the letter that was arbitrarily assigned to the side to which the dividing fence runs parallel, is:

• `L (bbx) = 3x + 2y`

It matters not that the farmer wishes to divide the area into 2 exact smaller areas.

Assuming the cost of the fencing is proportional to the length of fencing required, then:

• `C(bbx ) = alpha L(bbx)`

To optimise cost, using the Lagrange Multiplier `lambda`, with the area constraint :

• `bbnabla C(bbx) = lambda bbnabla A`

• `bbnabla L(bbx) = mu bbnabla A`

`implies mu = 3/y = 2/x implies x = 2/3 y`

• `implies xy = {(2/3 y^2),( 6*10^6 " sq ft"):}`

`:. qquad qquad {(y = 3*10^3 " ft"),(x = 2* 10^3 " ft"):}`

So the farmer minimises the cost by fencing-off in the ratio 2:3, either-way