# 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?

Jun 17, 2018

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

• $A \left(\boldsymbol{x}\right) = x y q \quad \left[= 6 \cdot {10}^{6} \text{ sq ft}\right]$

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:

• $L \left(\boldsymbol{x}\right) = 3 x + 2 y$

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:

• $C \left(\boldsymbol{x}\right) = \alpha L \left(\boldsymbol{x}\right)$

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

• $\boldsymbol{\nabla} C \left(\boldsymbol{x}\right) = \lambda \boldsymbol{\nabla} A$

• $\boldsymbol{\nabla} L \left(\boldsymbol{x}\right) = \mu \boldsymbol{\nabla} A$

$\implies \mu = \frac{3}{y} = \frac{2}{x} \implies x = \frac{2}{3} y$

• $\implies x y = \left\{\begin{matrix}\frac{2}{3} {y}^{2} \\ 6 \cdot {10}^{6} \text{ sq ft}\end{matrix}\right.$

$\therefore q \quad q \quad \left\{\begin{matrix}y = 3 \cdot {10}^{3} \text{ ft" \\ x = 2* 10^3 " ft}\end{matrix}\right.$

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