LAST EDITED ON Jul-28-03 AT 07:10 PM (EST)
As I had to learn English at an adult age, you have my sympathy. Your question does not make sense. Perhaps what you meant was"If the circumference (not edge!) of a certain rectangle is given, how long should the sides be to get the maximum area?"
Denote the rectangle sides x, y (x <= y). The circumference s and the area A are
s = 2x + 2y
A = xy
Since y = s/2 - x, the area is
A = x·(s/2 - x) = x·s/2 - x2
The area will be maximum, if the derivative dA/dx equals to zero.
dA/dx = d/dx{x·s/2 - x2} = s/2 - 2x = 0
x = s/4
y = s/2 - x = s/2 - s/4 = s/4
So that rectangle sides x = y = s/4 are equal and the rectangle is a square. But perhaps you are not familiar with calculus and the derivatives yet. Then look at the equation for the area A as a quadratic equation for x:
A = x·(s/2 - x) = x·s/2 - x2
x2 - x·s/2 + A = 0
The equation will have a real solution if its discriminant D is greater than or equal to zero:
D = (s/2)2 - 4A >= 0
4A <= (s/2)2
A <= (s/4)2
So the area A is always less than or equal to (s/4)2 and it will be maximum if it equals to (s/4)2. In that case the quadratic equation for x is
x2 - x·s/2 + (s/4)2 = 0
(x - s/4)2 = 0
x = s/4
y = s/2 - x = s/2 - s/4 = s/4
Again, the rectangle sides x = y = s/4 are equal and the rectangle is a square.
Regards, Vladimir