Answer:
 The box which minimizes the cost of materials has a square base of side length 4.16 cm and a height of 2.08 cm
Step-by-step explanation:
The cost is minimized when the cost of each pair of opposite sides is the same as the cost of the top and bottom. Since the top and bottom are half the cost of the sides (per unit area), the area of the square top and bottom will be double that of the sides. That is, the box is half as tall as wide, so is half of a cube of volume 72 cm³.
Each side of the square base is ∛72 = 2∛9 ≈ 4.16 cm. The height is half that, or 2.08 cm.
_____
If you want to see this analytically, you can write the equation for cost, using ...
 h = 36/s²
 cost = 2(1)(s²) + (2)(4s)(36/s²) = 2s² +288/s
The derivative is set to zero to minimize cost:
 d(cost)/ds = 4s -288/s² = 0
 s³ = 72 . . . . . multiply by s²/4
 s = ∛72 = 2∛9 ≈ 4.16 . . . . . cm
 h = 36/(2∛9)² = ∛9
The box is 2∛9 cm square and ∛9 cm high, about 4.16 cm square by 2.08 cm.
