Design a rectangular milk carton box of width w, length l, and height h which holds 496 cm3 of milk. The sides of the box cost 2 cent/cm2 and the top and bottom cost 4 cent/cm2. Find the dimensions of the box that minimize the total cost of materials used.

b = 8.12 cm

Explanation:

total cost = 2x3 (lb) + 2x1 (lh) + 2x1 (bh)

also lbh = 524

for minimum cost

total cost = 6h / 524 + 2 lh + 2 (524 / lh) (h)

= 6h / 524 + 2lh + 1024 / l

differentiating with respect to h,

0 = 0 + 2h - 1024 / l^2

so, h = 512 / l^2

similarly, b = 512 / l^2

hence, (512) ^2 / l^3 = 524

l = 7.94

so h = 512 / l^2 = 8.12 cm

and also b = 8.12 cm