Question

Please list out the steps to the problem?

An open rectangular box with a square base is to contain a volume of 18 cubic feet. The material for the base costs 8 cents per square foot. The material for the sides costs 6 cents per square foot. Find the dimensions of such a box that will make the cost of material a minimum.

Answer 1

Suppose that the length of the square base is `l` and the height of the box is `h`.

Since the volume is 18, `l^2h=18`, or `h=18/l^2`.

Now, we calculate the cost of the box. There is one square base costing `8` per unit area. Thus, the square base costs `8l^2`.

The `4` sides of the box costs `6` per unit area. Thus, the cost is `4*6*hl=24hl=432/l`.

We can find the total cost as `C=8l^2+432/l`.

Graphing this function gives us this:
graph{8x^2+432/x [-1,10,-100,1000]}

To find the minimum, we just need to find the `l` such that the slope of `C` is `0`.

Thus,
`(dC)/(dl)=d/(dl)(8l^2+432/l)`

`\ \ \ \ \ \ \ =16l-432/l^2`

Now, solve
`16l-432/l^2=0`

`16l=432/l^2`

`16l^3=432`

`l^3=432/16=27`

`l=3,h=18/l^2=2`

Thus, the dimensions of the box is `l=3,h=2`.