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?
1 Answer
Jun 17, 2018
If the rectangular field has notional sides
#A(bbx) = xy qquad [= 6*10^6 " sq ft"]#
The length of fencing required, if
#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

#bbnabla C(bbx) = lambda bbnabla A# 
#bbnabla L(bbx) = mu bbnabla A #
#implies xy = {(2/3 y^2),( 6*10^6 " sq ft"):} #
So the farmer minimises the cost by fencingoff in the ratio 2:3, eitherway