Imagine you are a manufacturer of cylindrical buckets. Yes, make the open bucket cylindrical to make life (the mathematics) a bit easier. You want the "bucket" to hold the maximum amount of water with the least amount of sheet metal. What would you think the height and radius of the bucket should be? Is there any special relation between its height and radius?

Note: This is a purely mathematical exercise. Do not worry about engineering aspects such as thickness of the materials or structural strength. We will leave such considerations to engineers.

please include a written explanation of each step