Find all possible values of the perimeter of a rectangle with sides $x, y \in \mathbb{Z}^+$, where its area is given by $A=3x+3y+\sqrt{9x^2+9y^2}$

Question

Find all possible values of the perimeter of a rectangle with sides positive integers $x$ and $y$ where its area is given by $A=3x+3y+\sqrt{9x^2+9y^2}$

In other words if $xy=3x+3y+\sqrt{9x^2+9y^2}$, find $2(x+y)$ where $x$ and $y$ are positive integers. I'm not sure how to start but maybe I can factor the expression out to make it $3(x+y+\sqrt{x^2+y^2})$ but I can't go any further. Any help would be much appreciated!

Answer 1

This is an application based calculus question. You have find the total $(x , y)$ values suitable under the area/perimeter and +integer restraint. By conventional methods, you should begin by expressing $y$ in terms of $x$ to obtain a function of the rectangle's perimeter under a single variable. This eases the next step of differentiating to evaluate the maximum $\lfloor{x}\rfloor$ of this perimeter function such that both $x$ and the perimeter are positive. Finally, find all the equivalent $y$-values and eliminate any pair of $(x, y)$ that are not positive integers.

Here is a little help on the factoring to get you started:

In cases where you have the unknown variable under a root and outside the root, you should first try to remove/eliminate the root term:

$$xy = 3x + 3y + \sqrt{9x^2 + 9y^2} \\\iff \sqrt{9x^2 + 9y^2} = xy - 3x - 3y \\\iff 9x^2 + 9y^2 = (xy - 3x - 3y)^2 = 9x^2 + 9y^2 + x^2y^2 - 6x^2y - 6xy^2 + 18xy \\\iff x^2y^2 - 6x^2y - 6xy^2 + 18xy = 0$$

Can you continue to factor from here?