The equations for perimeter and area for your problem are:
a.) Write the formula for area in terms of L.
We know that 100 ft of fencing is available to make the dog pen, so this will be the length of the perimeter:
Now plug this into the area equation and a substitution for W, so your area will be in terms of L.
b.) Determine the dimensions of the pen that will maximize the area.
To find the maximum area that can be created take the derivative of your new area equation.
To find the maximum length, set your equation equal to zero:
50 - 2L = 0
Solve for L. This will be the max length of your dog pen. Then you can plug the value for L into your perimeter equation to find out what W is and then go from there to get your max area.
c.) Suppose there is already a wall in place for one side of the pen. If the same amount of fencing is to be used, what dimensions will maximize the area of the pen?
Look back at our perimeter equation. This part will alter that equation to account for the fact that there are only 3 sides we need to fence in. So, can you think of how you might alter that equation to get the answer for this last one?