Covering a marble Imagine a flat-bottomed cylindrical pot with a circular cross section of radius 4. A marble with radius 0 < r < 4 is placed in the bottom of the pot. What is the radius of the marble that requires the most water to cover it completely?

Verified step by step guidance

1. Understand that the problem involves finding the radius of a marble that maximizes the volume of water needed to cover it in a cylindrical pot.
2. Recognize that the volume of water required to cover the marble is related to the height of the water above the marble, which depends on the radius of the marble and the geometry of the pot.
3. Set up the relationship between the radius of the marble (r) and the height of the water (h) in the pot, using the formula for the volume of a cylinder: V = πR²h, where R is the radius of the pot.
4. Express the height of the water in terms of the radius of the marble, considering that the water will fill the space above the marble to the top of the pot.
5. Differentiate the volume function with respect to the radius of the marble and find the critical points to determine the radius that maximizes the volume of water needed.