What is the volume of the largest right cylinder that can fit inside a closed rectangular box measuring $12$ inches by $10$ inches by $8$ inches?
I thought we assume the radius of the cylinder equals to half the maximum dimension in the rectangular prism. $r=\dfrac{1}{2}max(\{12,10,8\},h=min(2\cdot r,\{10,12,8\} \cap \{2r\}^C)$
I am also not sure what the cross-section should look if were not to enclose the cylinder in a sphere.
I imagine we center the cylinder's circular cross-section on the middle of the rectangle, but I am not sure on what rectangular side is most optimal.
I am not sure whether $12$ refers to a length, width, or height.