Question 1048501
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I found a solution to this, although it is decidedly inelegant.  Nevertheless, it is an answer.  I used your equation, but moved the constant to the other side


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 8x\ +\ 8y\ =\ xy\ -\ 1]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 8x\ -\ xy\ +\ 8y\ =\ -1]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ y(8\ -\ x)\ =\ -8x\ -\ 1]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ y\ =\ \frac{-8x\ -\ 1}{8\ -\ x}]


Now, note that for all positive integer values of *[tex \Large x], the numerator in the right-hand side is negative.  For all integer values less than 8, the denominator is positive and the value of *[tex \Large y] is negative which is absurd.  If *[tex \Large x\ =\ 8], *[tex \Large y] is undefined.


If you graph the function *[tex \Large y\ =\ \frac{-8x\ -\ 1}{8\ -\ x}] on the domain *[tex \Large 9\ \leq\ x\ <\ \infty], you note that the graph has a horizontal asymptote at 8.


A bit of fiddling with Excel yields the following:


For *[tex \Large x\ =\ 9], *[tex \Large y\ =\ 73], and the perimeter is 164.


For *[tex \Large x\ =\ 13], *[tex \Large y\ =\ 21], and the perimeter is 68.


For *[tex \Large x\ =\ 21], *[tex \Large y\ =\ 13], and the perimeter is 68.


And for *[tex \Large x\ =\ 73], *[tex \Large y\ =\ 9], and the perimeter is 164.


And there are no other integer solutions to the equation because when *[tex \Large x\ >\ 73], the value of the function begins to approach the value 8 to which the function is asymptotic.  Hence, your maximum perimeter for integer dimensions is 164.


John
*[tex \LARGE e^{i\pi}\ +\ 1\ =\ 0]
My calculator said it, I believe it, that settles it
<img src="http://c0rk.blogs.com/gr0undzer0/darwin-fish.jpg">
*[tex \Large \ \
*[tex \LARGE \ \ \ \ \ \ \ \ \ \  

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