A factory builds an open-top storage tank: a rectangular box with a square base, four vertical walls, and no lid. It must hold exactly cubic metres.
The base is reinforced and costs more per square metre than the walls do:
| Part of the tank | Cost per square metre |
|---|---|
| The square base | 5 dollars |
| Each of the four walls | 3 dollars |
Let be the side of the square base and the height, both in metres.
Task: choose and so that the tank holds exactly cubic metres at the lowest possible total material cost, then report that minimum total cost in dollars.
Give an exact integer number of dollars. (Confirm your critical point really is a minimum before you report it.)