SOLUTION: A rectangular box is to have a square base and a volume of 36 feet cubed. If the material for the base costs $.18/square foot, the material for the sides costs $.12/square foot, an

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Question 1167566: A rectangular box is to have a square base and a volume of 36 feet cubed. If the material for the base costs $.18/square foot, the material for the sides costs $.12/square foot, and the material for the top costs $.14/square foot, determine the dimensions of the box that can be constructed at minimun cost.
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


Let the square base have side x.

Then, since the volume is 36, the height of the box is 36/x^2.

The top and bottom each have area x^2; each side has area x(36/x^2) = 36/x.

The cost is 18 cents per unit area for the bottom, 12 cents per unit area for the four sides, and 14 cents per unit area for the top. The total cost is

32%28x%5E2%29%2B4%2812%29%2836%2Fx%29+=+32x%5E2%2B1728%2Fx

Use calculus (tedious) or a graphing calculator (fast) to find that the minimum cost is when x=3; then use that to answer the question.