An open box is to be made from a square piece of material 12 inches on a side by cutting equal squares of length x from the corners and turning up the sides.
(a)Write the volume (V) as a function of x.

(b) What is the domain of this function?

(Hint: Solve 0 < 12 – 2 x < 12)

2. $\displaystyle V=x(12 - 2x)^2$

3. If you are looking for the practical domain of $\displaystyle V(x) = x(12-2x)^2$, since $\displaystyle (12-2x)^2$ must always be positive, if we have $\displaystyle x \leq 0$, then $\displaystyle x(12-2x)^2$ will not be positive, but since Volume is positive, we must have $\displaystyle x> 0$. So the domain is $\displaystyle x \in R^{+}$

4. Originally Posted by DivideBy0
..., since $\displaystyle (12-2x)^2$ must always be positive, ...
Hello,

this statement isn't quite correct. With this problem it is necessary that

$\displaystyle 12-2x \geq 0~\wedge~x \geq 0$

Otherwise you cut off more than there exist - and that means you build a box by cutting off parts of the 3-D-space to get a negative space