Question 957901
Total perimeter (3 sides only): 32+4= 36 (32 of railing + 4ft opening for stairs)
I’ll call the Length X and the Width Y:
Maximize A = area of the rectangle = (Length)(Width) = X*Y
given a perimeter of X+2Y= 36 (only 1 X, the other is against the bungalow-see my illustration)
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Now we need to solve for one of the variables.  Since  X+2Y= 36, 
2Y= 36-X; Y= (36-X)/2;  Y= 18- X/2
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Now that we solved for Y, we have that:
A= X*Y= X*(18- X/2)= 18X-X^2/2
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We want to maximize A:
A’= 18-X. The only time A’=0 is when X= 18. 
So, 0<=X<=36. Why? Because we make it such that the only critical points of A are when X=0, 18, and 36, and the maximum area will be given by one of them.
Here we go:
At the critical number X= 0: A= 18(0) - (0)^2/2= 0 square feet – toss this answer out.
At the critical number X= 18: A= 18(18) – (18)^2/2 = 162
At the critical number X= 36: A= 18(36) – (36)^2/2 = 0 – toss this one out, too.

So our largest possible area is 162 square feet and the dimensions are: length (X)= 18 feet  and width (Y)= 18 - X/2= 18 - 9= 9 feet.
Check: 18 + 2(9)= 36 feet, 32 of railing+4 feet opening on the railing for the stairs.
*[illustration Railing.JPG]