A rectangular box with no top must have a volume of 2,500 cm³.

Three different materials will be used in the construction of this box. The bottom will be made out of material that costs 5 cents/cm², the front and back will be made out of material that costs 4 cents/cm², and material for the two sides costs 2 cents/cm².

What are the dimensions of the box (with volume 2,500 cm³) that will minimize the total cost of the materials?

After having done a number of problems of this type, I realized there was a generalization that could be made on minimizing the cost for a given volume: the total cost of top and bottom should equal the total cost of front and back and also equal the total cost of the left and right sides.

Let d, w and h be the depth, width and height respectively.

5 dw = 8 hw = 4 dh

(two of the dimension-pairs' costs have been doubled from the material costs as opposite faces are considered together; the bottom is not matched with a top, however, so the 5 cents / cm^2 stands.)

5 dw = 8 hw
5 d = 8 h

8 hw = 4 dh
8 w = 4 d
2 w = d

5 dw = 4 dh
5 w = 4 h

also, wdh = 2500

Converting the last to just w:

w(2w)(5w/4) = 2500
10 w^3 / 4 = 2500
w^3 = 1000
w = 10
d = 20
h = 50/4 = 12.5

The check below of the vicinity (w and d at the given values surrounded by +/- 1 mm) of the answer in terms of dollar cost shows indeed a local minimum of cost here:

 30.00176   30.00101   30.00076
 30.00025   30.00000   30.00025
 30.00074   30.00099   30.00174

DEFDBL A-Z
FOR w = 9.9# TO 10.11# STEP .1#
  FOR d = 19.9# TO 20.11# STEP .1#
    h = 2500 / (w * d)
    PRINT USING "#####.#####"; (5 * w * d + 8 * w * h + 4 * d * h) / 100;
  NEXT d
  PRINT
NEXT w
PRINT

So the box is 10 cm wide, 20 cm deep and 12.5 cm high.

Posted by Charlie on 2009-05-15 14:09:30