A farmer wishes to build a fence for 6 adjacent rectangular pens. If there is 600 feet of fencing available, what are the dimensions of each pen that maximizes total pen area?
The image looks like this:
box box box
box box box
also:
If the interior fencing is $3.00 per foot and the perimeter is $5.00 per foot, what are the pen dimensions that minimize cost?
3 answers
assume each pen encloses 200 square feet of area in the 2nd part sorry!
part 2:
if each pen has width x and height y
xy=200
6x+8y=600
you have no room to vary the dimensions. So, I assume the 600 feet of fencing does not apply to part 2. Accordingly,
xy=200
cost c = 5(3x+4y)+3(3x+4y)
= 24x+32y
= 24x+32(200/x)
dc/dx = 24 - 6400/x^2
= 8(3x^2-800)/x^2
dc/dx = 0 when x = 20√(2/3)
so, the pens are 20√(2/3) by 10√(3/2)
if each pen has width x and height y
xy=200
6x+8y=600
you have no room to vary the dimensions. So, I assume the 600 feet of fencing does not apply to part 2. Accordingly,
xy=200
cost c = 5(3x+4y)+3(3x+4y)
= 24x+32y
= 24x+32(200/x)
dc/dx = 24 - 6400/x^2
= 8(3x^2-800)/x^2
dc/dx = 0 when x = 20√(2/3)
so, the pens are 20√(2/3) by 10√(3/2)
25'x25'
2 answers