Afarmer wants to fence an area of 37.5 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. what should the lengths of the sides of the rectangular field be so as to minimize the cost of the fence

The fence cost will be optimized when the total length of fence in one direction (2 lengths of fence) is equal to the total length of fence in the other direction (3 lengths of fence). In order for that to be the case, the aspect ratio of the field must be 3:2. The short side will be √(2/3*37.5*10^6 ft²) = 5000 ft.

The long side will be 7500 ft, the short sides and the center partition will be 5000 ft each.

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Suppose the side of the field perpendicular to the partition is length x. Then the total length of fencing is
.. length = 2x + 3*(37500000/x)
This has a minimum at x=7500 ft. The corresponding total length of fence is 30,000 ft.