a farmer has 360 feet of fencing to make three identical adjacent rectangular pens. What dimensions of each pen will maximize the total enclosed area?

Answer 1

so if each pen has dimensions x and y, we have
3x+4y+3x = 360
6x+4y = 360
3x+2y = 180
y = 90 - 3/2 x
the area is
a = 3xy = 3x(90 - 3/2 x) = 270x - 9/2 x^2
da/dx = 270 - 9x
da/dx = 0 when x = 30
So each pen is 30 by 45

As is always the case in such problems, maximum area is achieved when the total fencing is divided equally among lengths and widths.
360/2 = 180
x = 180/6 = 30
y = 180/4 = 45