Question

Suppose you want to fence in a field,
and then subdivide it into three equal parts.
You have 2400 yards of fencing material to
use. What should be the dimensions of one
of the three smaller fields, if you want to
fence in the maximum area?

Answer 1

I will assume that the three smaller parts are equal rectangles.
Let the width of one of the smaller rectangles is x
and its length is y
Make a sketch and you should see that
our original field is 3x by y
and perimeter is 6x + 4y
so 6x + 4y = 2400
3x + 2y = 1200
y = (1200 - 3x)/2 = 600 - 3x/2

area = 3xy
= 3x(600 - 3x/2)
= 1800x - (9/2)x^2

d(area)/dx = 1800 - 9x
= 0 for a max of area
9x = 1800
x = 200 and y = 600 - 3(200)/2 = 300

So each of the smaller fields is 200 by 300 yards

Answer 2

Note that the maximum area is achieved when the fencing is divided equally among the lengths and widths: 1200 each.