A jewelry box with a square base is to be built with silver plated sides, nickel plated bottom and top, and a volume of 36 cm3. If nickel plating costs $1 per cm2 and silver plating costs $2 per cm2, find the dimensions of the box to minimize the cost of the materials. (Round your answers to two decimal places.) The box which minimizes the cost of materials has a square base of side length _ cm and a height of__ cm

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.