A box with an open top is made from a square piece of cardboard, of which side length is 100 cm, by cutting a square from each corner and then folding up the sides. Find the dimensions of the box of the largest volume.

I drew a diagram and on any given side of 100 cm your going to have two pieces being cut away so I made the sum of the pieces and part left over

$\displaystyle 2x + y = 100 \Rightarrow y=100-2x \Rightarrow 2x+(100-2x) = 100$

$\displaystyle V=y^3$

and so

$\displaystyle V(x) = [100-2x]^3$ and $\displaystyle V'(x) = -6(100-2x)^2$

solving $\displaystyle V'(x)=0, x = 50$

my interval is $\displaystyle 0 \le x \le 100$

Is this correct so far? because if x = 50 than 2x=100 which doesn't seem right to me