Question
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?
Answers
oobleck
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
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