Design a rectangular milk carton box (PLS FAST İ HAVE 1 HOUR)

Question

Design a rectangular milk carton box of width w, length l, and height h which holds 532 cm3 of milk. The sides of the box cost 1 cent/cm2 and the top and bottom cost 2 cent/cm2. Find the dimensions of the box that minimize the total cost of materials used.

Answer 1

We have four sides of the box, 2 broad sides with areas height x length, 2 narrow sides with areas height x width, and the top and bottom with areas width x length.

So for the cost of the box f(w,l,h) = 2hl + 2hw + 2(2wl) [the sides are 1 cent/cm²; top and bottom are 2 cent/cm²]

We have the cost f(w,l,h) subject to the contstraint of the Volume V(w,l,h)=wlh=532 cm³

The Lagrange condition is ∇f=λ∇V and V(w,l,h)=532. This gives equations:

f_w=λlh, f_l=λwh, f_h=λwl.