SOLUTION: I'm not even sure if this is the right topic, but here's my problem. The lengths of the sides of a rectangle are all integers. Four times its perimeter is numerically equal to one

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Question 1048501: I'm not even sure if this is the right topic, but here's my problem.
The lengths of the sides of a rectangle are all integers. Four times its perimeter is numerically equal to one less than its area. Find the largest possible perimeter of such a rectangle.
Now I tried manipulating the variables but I'm stuck here.
8x+8y+1=xy
I don't know how to relate this with the maximum value or solve x and y independently or like that, so I really need help.
Thanks.
Found 2 solutions by josgarithmetic, solver91311:
Answer by josgarithmetic(39616) About Me  (Show Source):
You can put this solution on YOUR website!
x and y are expected to be integers (or better, whole numbers).
4%282x%2B2y%29=xy-1----------the description.

The perimeter, as a function may be highlight_green%28p=2x%2B2y%29, and you would want to be able to substitute for either x or for y.

Look again at the equation formed from the description.
8x%2B8y=xy-1
8x%2B8y-xy=-1
8y-xy=-1-8x
y%288-x%29=-1-8x
y=%28-8x-1%29%2F%288-x%29
y=%28%28-8x-1%29%2F%288-x%29%29%28%28-1%29%2F%28-1%29%29------multiplying by 1 to produce more convenient formula,
highlight_green%28y=%288x-1%29%2F%28x-8%29%29


I leave the rest of this problem unfinished so that you could continue and finish.

Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!


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









Now, note that for all positive integer values of , the numerator in the right-hand side is negative. For all integer values less than 8, the denominator is positive and the value of is negative which is absurd. If , is undefined.

If you graph the function on the domain , you note that the graph has a horizontal asymptote at 8.

A bit of fiddling with Excel yields the following:

For , , and the perimeter is 164.

For , , and the perimeter is 68.

For , , and the perimeter is 68.

And for , , and the perimeter is 164.

And there are no other integer solutions to the equation because when , 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

My calculator said it, I believe it, that settles it