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?

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Answer 1 · 2018-06-17T23:09:30

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

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:

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:

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

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

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

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