Max is building a rectangular dog pen. one side of the pen will just be the side of the house. if he has 45 feet of fencing to use for the other three sides, what is the largest area he can create? round to the nearest tenth.

Let the shorter sides of the rectangle be x feet 
then the length of the 3 sides will be x , x and  45 - 2x

The area of the rectangle  will be x(45-2x) =  45x - 2x^2 
we need to find the maximum value of this area

Find the derivative:-

A'  =  45 - 4x  = 0 for a maximum values

4x = 45 

x =  45/4 = 11.25

So maximum area = 11.25(45 - 22.5) =  253.1  square feet