Question

You have 700 ft of fencing to make a pen for hogs. If you have a river on one side of your property, what are the dimensions (in ft) of the rectangular pen that maximize the area?
shorter side

longer side

Answer 1

I suggest you try different dimensions. Which will give you the greatest area?

400 feet along the river; 400 feet of fencing parallel to the river, 150 feet of fencing at each end
or
600 feet along the river, 600 feet of fencing parallel to the river, 10 feet of fencing at each end
or
?

Answer 2

If you play around with it, you will see that the maximum area in such problems is always attained when the fencing is divided equally among lengths and widths. Here, you have one length and two widths, so if 350 ft of fencing is allocated to each set, then the pen will be

350 x 175

for maximum area

Answer 3

I think I can mathematically prove it using L'Hopitals rule.