A closed box with a square base is to have a volume of 16,000 cubic cm. THe mate

Question

A closed box with a square base is to have a volume of 16,000 cubic cm. THe material for the top and bottom of the box costs $3. per square cm and the material for the sides costs $1.50 per square cm. Let x be the length of one edge of the base of the box, in cm.
Let C denote the cost, in dollars, of material for the box. Write a formula for C in terms of only x and determine the dimensions of the cheapest box. You need to use the appropriate derivative function to locate critical points for the cost function. State your conclusion with appropriate labels and show your work.

Explanation / Answer

h = height

V = x2h = 16000 -> h = 16000/x2

C(x) = 3*(2x2) + 1.5*(4xh) = 6x2 + 6xh = 6x2 + 96000/x

To find minimum of C(x) we have:

C'(x) = 12x - 96000/x2 = (12x3 - 96000)/x2 = 12(x3 - 8000)/x2 = 0

x3 = 8000 -> x = 20   -> h = 16000/202 = 40

So the dimensions for minimum cost are:

20, 20, 40

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