Find the dimensions of the field that maximizes the total area

A rectangular field is to be subdivided in 2 equal fields. There is 8350 feet of fencing available.

Find the dimensions of the field that maximizes the total area. (List the longer side first)

Width =_____________ feet

Length =________________ feet

What is the maximum area ? Area =_________________

Re: Find the dimensions of the field that maximizes the total area

width = w, length = l, P = 8350 (perimeter), A = total area. Units are everywhere feet.

P = 2w + 3l (That's 2w + 2l to bound the area, and another fence of length l down the middle to split the fields.)

Thus l = (P-2w)/3.

Note w>0, l>0. Thus also (P-2w)/3 >0, so w < P/2.

A = lw = w(P-2w)/3.

Translation: Maximize A(w) = w(P-2w)/3 where 0<w<P/2, and where P is constant.

Re: Find the dimensions of the field that maximizes the total area

We have , and .

By AM-GM inequality,

The equality case occurs when 2w = 3l.

Re: Find the dimensions of the field that maximizes the total area