This is an optimization problem, so lets start by setting up a function with one variable and taking a derivative. Then we can find any zeros of that derivative.
The problem states that we have 3 parallel partitions, so lets start with a basic drawing.
In the drawing above we have
Additionally, we are looking for the area, so we will need the formula for the area of a rectangle:
Now we have two equations that we can combine to get one function to take the derivative of. Lets solve the first equation for
Now we can replace
To make the derivative easier, lets make this a polynomial.
We are looking for the extrema, which occur at turning points or where the derivative is equal to 0. Lets use the power rule to get the derivative and set it equal to 0.
Great, we now have the length of one side of our fence! Lets use it to solve for
Now we have the dimensions for our fence. The largest possible pen would have a length of