You must design a closed rectangular box of width w, length l and height h, whose volume is 504 cm3. the sides of the box cost 3 cents/cm2 and the top and bottom cost 4 cents/cm2. find the dimensions of the box that minimize the total cost of the materials used.

Dimensions will be

Length = 7.23 cm

Width = 7.23 cm

Height = 9.64 cm

Step-by-step explanation:

A closed box has length = l cm

width of the box = w cm

height of the box = h cm

Volume of the rectangular box = lwh

504 = lwh

Sides which involve length and width and height, cost = 3 cents per cm²

Top and bottom of the box costs = 4 cents per cm²

Cost of the sides = 3[2(l + w)h] = 6(l + w)h

= 3[2(l + w)h]

Cost of the top and the bottom = 4(2lw) = 8lw

Total cost of the box C = + 8lw

                                      = + 8lw

To minimize the cost of the sides

---------(1)

-------(2)

Now place the value of w from equation (1) to equation (2)

l³ = 378

l = ∛378 = 7.23 cm

From equation (2)

w = 7.23 cm

As lwh = 504 cm³

(7.23)²h = 504

h = 9.64 cm