Question

You have $1200 to buy fencing, which cost $10 per horizontal foot. The fence will surround a rectangular field and split down the middle. What are the dimensions that will enclose the maximum possible area?

Answer 1

It seems that you can buy a maximum of 120 feet of fencing with your budget. Your rectangle will have a length of a and a width of b. The rectangle itself will have a perimeter of 2a+2b. You also have a split down the middle, another length of a. The total amount of fencing you need is 3a+2b. The area of your field is a*b. So...

3a+2b=120
a*b need to be maximized.

2b=120-3a
b=60-1.5a

a(60-1.5a)
60a-1.5a^2
maximize by setting the derivative equal to 0
60-3a=0
60=3a
a=20

3(20)+2b=120
2b=60
b=30

The dimensions are 20x30.