Saturday

April 18, 2015

April 18, 2015

Posted by **Michael** on Monday, March 26, 2012 at 4:31pm.

In the example they wish to describe, it is assumed that the farmer has 900 feet of fencing with which to erect a rectangular pen alongside a long, existing fence (so the existing fence forms one side of the pen). Suppose the pen is to be subdivided into four parts in a two-by-two arrangement by including interior fences parallel to the outside boundaries. Then what dimensions make for the largest combined area? What if the farmer subdivides into nine pens in a three-by-three arrangement? What if the farmer subdivides into n2 pens in an n-by-n arrangement?

- Calculus -
**bobpursley**, Monday, March 26, 2012 at 4:41pmI don't understand the two by two arrangement.

What you need to do is write an expression for the total amount of fence (outside+inside lengths)=900

it will be in terms of fractions of length, width.

Then you want to maximize area:

Area= Outsidelength*width.

Now in the fence length needed equation, solve for w. Put that in the area equation.

Take the derivative dA/dl, set to zero, and solve for l.

- Calculus -
**Steve**, Monday, March 26, 2012 at 4:46pmMight as well go for the general nxn case, then we can plug in what we want.

If there are n^2 small pens, with width x and length y, with length parallel to the long fence, then the total fence used is

n(n+1)x + n^2y = 900

y = (900-n(n+1)x)/n^2

A = n^2 xy

= n^2 x (900-n(n+1)x)/n^2

= x(900-n(n+1)x)

= 900x - n(n+1)x^2

dA/dx = 900 - 2n(n+1)x

max at x = 900/[2n(n+1)]

x = 450/[n(n+1)]

y = 450/n^2

**Answer this Question**

**Related Questions**

Calculus - The Regional Farm Bureau (RFB) is preparing a brochure that offers ...

math - A farmer has both chickens and sheep on his farm. He has a total of 25 ...

math - In a farm the ratio of goats to cows is 3:4 if there are 420 farm animals...

math - A farmer has $100.he has to buy 100 farm animals.He has to spend exactly...

english - Of all of the characters in Animal Farm, are there any who seem to ...

Animal Farm - Has anyone read the book Animal Farm by George Orwell? If so, they...

Maths - A farm rears 9 ducks and cows in his farm. There are 28 legs altogether....

Maths - A farm rears 9 ducks and cows in his farm. There are 28 legs altogether....

Math - On joes farm there are 99 animals. There are four times as many cows as ...

algebra - A farm raises a total of 220 chickens and pigs. The number of legs of ...