A bakery is designing a custom cake box using a $20$-inch by $30$-inch sheet of cardboard. The cardboard is folded in half to form a $20$-inch by $15$-inch rectangle. To create the sides of the box, four identical squares with side length $x$ inches are cut from each corner of the folded rectangle. The cardboard is then unfolded, and the six flaps formed by the cuts are folded upward to create the sides and lid of the box. Let $V\left(x\right)$ represent the volume of the finished cake box in terms of $x$. Determine the domain of $V$.