A farmer wants to fence an area of 24 million sqft in a rect field and divide in half with fence parallel to one of the sides.?

Starting with the area of a rectangle:

A = lw

We know the area is 24,000,000 ft²:

24000000 = lw

Presuming the cost of all parts of the fence is the same, and we want the fence to cut the area in half including be around it, then we have 3 lengths and 2 widths (vs. the normal 2 of each).  The amount of fencing needed is then:

f(l, w) = 3l + 2w

If we solve the first equation for l in terms of w, we can substitute to get a function for the amount of fencing needed in terms of only width:

24000000 = lw

24000000 / w = l

f(l, w) = 3l + 2w

f(w) = 3(24000000 / w) + 2w

f(w) = 72000000 / w + 2w

We can find the "w" that gives a minimum f(w) by solving for the zero of the first derivative:

f'(w) = -72000000 / w² + 2

0 = -72000000 / w² + 2

Multiply both sides by w²:

0 = -72000000 + 2w²

72000000 = 2w²

36000000 = w²

w = ± 6000

We can't have a negative width, so throw that out to be:

w = 6000 ft

Now we can solve for l:

l = 24000000 / w

l = 24000000 / 6000

l = 4000 ft

The shorter side is 4000 ft and the longer side is 6000 ft.