Posted by **Henry** on Wednesday, April 10, 2013 at 1:12pm.

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?

I keep getting x=150 but I have been told that is not enough fencing. Can anyone help?

- Calculus 2 -
**Reiny**, Wednesday, April 10, 2013 at 1:18pm
You must give a description of the pens.

Is there a large rectangle with equal partitions parallel to the widths ? (the usual case)

Since you don't say what the x stood for, I have no way of telling what the 150 represents, since "dimension" implies length and width.

- Calculus 2 -
**Henry**, Wednesday, April 10, 2013 at 1:28pm
That was the only thing the question said :/. It never gave a description of the pens. There is a picture with 6 boxes connected to each other 3 boxes on top and 3 on bottom:

box box box

box box box

other than that, that was all the info I was given:(

- Calculus 2 -
**Steve**, Wednesday, April 10, 2013 at 3:15pm
In that case, if each pen has width x and height y in the drawing, then

total area is 6xy

Also, 3x+3x+3x+2y+2y+2y+2y = 600, so 9x+8y=600

a = 6xy = 6x(600-9x)/8

= 9/4 x(200-3x)

da/dx = 9/2 (100-3x)

so, da/dx = 0 when x = 100/3

so, each small pen is 100/3 by 75/2

max area = 7500

- Calculus 2 -
**Reiny**, Wednesday, April 10, 2013 at 3:56pm
Ok, then it isn't that bad

Make a sketch,

label the length of each small pen as x and its width y

counting up all the x's and y's, I get

9x + 8y = 600

y = (600 - 9x)/8

where 600-9x > 0

9x < 600

x < 66.67

area = 3x(2y)= 6xy

= 6x(600-9x)/8

= 3600x - (27/4)x^2

This is a parabola which opens downwards, so it has a maximum

the x of the vertex is -b/(2a) = -3600/(-27/2) =266.67

which is beyond our restriction of x

Thus this question has no solution

by Calculus:

d(area)/dx = 3600 - 27x/2

= 0 for a max area

27x/2 = 3600

27x = 7200

x = 266.66..

y = (600 - 9(266.67)/8 which is a negative

no solution

here is a picture of why

http://www.wolframalpha.com/input/?i=plot+y+%3D+3600x+-+%2827%2F4%29x%5E2

- Just ignore my last post -Calculus 2 -
**Reiny**, Wednesday, April 10, 2013 at 4:00pm
What stupid error that was.

my mistake is in

= 6x(600-9x)/8

= 3600x - (27/4)x^2

what garbage that is !!!

Go with STeve

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