a farmer wants to make a rectangular pen from 200 feet of fencing. he plans to use a 100 foot wall along with some of the fencing to make one side of the pen. find the dimensions of the pen that will make the enclosed area as large as possible.

presumably, he wants to use all of the fencing, if he wants as large a pen as possible...

2x+y=200
a = xy = x(200-2x) = 200x-2x^2
da/dx = 200-4x
da/dx=0 when x=50

the pen is 50x100

In this kind of problem, the fencing is always divded equally among the lengths and widths.

To find the dimensions of the pen that will make the enclosed area as large as possible, we can use calculus and optimization techniques.

Let's assume the length of the pen, parallel to the 100-foot wall, is x feet. In this case, one side of the pen will be the 100-foot wall, and the other three sides will be made up of x feet of fencing.

The perimeter of the pen is given by:
Perimeter = 100 + 2x

Since the total amount of fencing available is 200 feet, we can use this to set up an equation:
100 + 2x + x + x = 200

Simplifying the equation, we have:
4x + 100 = 200
4x = 100
x = 25

So the length of the pen, parallel to the 100-foot wall, is 25 feet. We can also find the width of the pen by subtracting twice the length from the remaining fencing:
Width = (200 - (100 + 2x))/2
Width = (200 - (100 + 2(25)))/2
Width = (200 - (100 + 50))/2
Width = (200 - 150)/2
Width = 50/2
Width = 25

Therefore, the dimensions of the pen that will make the enclosed area as large as possible are 25 feet by 25 feet.