Q:

Finding the length and width of a house that maximize its area

A house is built in the shape of a rectangle, with $3$ rectangular interior sections separated by parallel walls, using fencing. The owner has $900$ feet of fencing, and he wants to enclose the largest possible area. What should the length, width, and area be? 

Please help, I'm lost.

A:

Let the two inside parallel walls each have length $x$. Let the sides of the rectangle perpendicular to these each have length $y$. 
Then the total area enclosed is $xy$. The amount of fencing used is $4x+2y$.  This is to be $900$, since it is clear that it is best to use up all the fencing.
So we want to maximize $xy$, under the constraint $4x+2y=900$.  
Thus $y=450-2x$, and we want to maximize $x(450-2x)$.
Because of the physical situation, we need $x\ge 0$ and $y\ge 0$. This means $x\le 225$.
So mathematically, we want to minimize $f(x)=450x-2x^2$, where $0\le x\le 225$.
This can be done by standard tools, such as calculus or completing the square.

