Optimal popcorn box A small popcorn box is created from a 12" x 12" sheet of paperboard by first cutting out four shaded rectangles, each of length x and width x/2 (see figure). The remaining paperboard is folded along the solid lines to form a box. What dimensions of the box maximize the volume of the box? <IMAGE>

Verified step by step guidance

1. Define the dimensions of the box after cutting out the corners: the length will be (12 - x) inches, the width will be (12 - x) inches, and the height will be x/2 inches.
2. Write the volume V of the box as a function of x: V(x) = (12 - x)(12 - x)(x/2).
3. Simplify the volume function: V(x) = (12 - x)^2 * (x/2) = (1/2)(12 - x)^2 * x.
4. Find the critical points by taking the derivative of V with respect to x and setting it equal to zero: V'(x) = 0.
5. Determine the maximum volume by analyzing the critical points and endpoints of the interval for x, which is from 0 to 12.