Partial Derivatives

Question

Design a rectangular milk carton box of width w, length l, and height h which holds 516 cm³ of milk. The sides of the box cost 3 cent/cm² and the top and bottom cost 5 cent/cm2². Find the dimensions of the box that minimize the total cost of materials used.

dimensions =

Answer 1

Let w=width, h= height and L=length

Volume, V = 516 = whL --> L= 516/wh

Cost, C = (0.03)(2)(wh+hL)+ (0.05)(2)wL

C(w,h) = 0.06wh+0.06h(516/(wh))+0.1w(516/(wh))

= 0.06wh+30.96/w+51.6/h

∂C/∂w = 0.06h-30.96/w²

∂C/∂h = 0.06w-51.6/h²

Setting each partial derivative to zero and manipulating the equations:

h = 516/w²

wh² = 860

w = (516²/860)^1/3 ≈ 6.76 cm

h ≈ 11.27 cm

L ≈ 6.77 cm