A box is with a square base and open top is to be constructed and a total volume of 720 cubic inches is required. The cost of material for the base is 8 dollars per square inch and the cost of material for the sides is 6 dollars per square inch. Express the total cost of the box as a function of the length of the base.

  total cost = 8x^2 +17280/x

Step-by-step explanation:

Let x represent the base length. Then the area of the base is x^2, and the height is h = 720/x^2.

The area of the four sides is ...

  (4x)(h) = (4x)(720/x^2) = 2880/x

The cost of the base is ...

  base cost = 8x^2

And the cost of the sides is ...

  side cost = 6(2880)/x = 17280/x

The total cost of the box is ...

  total cost = base cost + side cost

  total cost = 8x^2 +17280/x

Comment on the cost function

You will find this function has a minimum at x=∛1080 ≈ 10.260 in. The total cost is about $2526.35, and the box is 2/3 times as tall as wide. That aspect ratio makes any pair of opposite sides cost the same as the base, the generic solution to a cost optimization problem of this sort.