How do you find the dimensions of the box that minimize the total cost of materials used if a rectangular milk carton box of width w, length l, and height h holds 534 cubic cm of milk and the sides of the box cost 4 cents per square cm and the top and bottom cost 8 cents per square cm?
Note that varying the length and width to be other than equal reduces the volume for the same total (length + width); or, stated another way,
Using given information about the Volume, express the height (
Write an expression for the Cost in terms of only the width (
Take the derivative of the Cost with respect to width and set it to zero to determine critical point(s).
Cost = (Cost of sides) + (Cost of top and bottom)
and with some simple numeric division: