For simplicity, the cone and cylinder are shown using a single side view - the cylinder shown in dark red. Choosing the cone's vertex as origin, and given cone diameter and lateral edge length at 10 and 13 respectively, the altitude is deduced via Pythagoras at 12, and the diagram is thus explained. The volume of a right cylinder comes by product of its base area and height. If you use the values diagramed below, the volume is easily expressed as a function of radius, "r sub-c". Finally, differentiate the volume function, V, and set resulting derivitive equal to zero, i.e., let V' = 0 (why?). Solve V' = 0 for r sub-c and you'll have the radius of the cylinder whose volume exceedes all others. I shall assume you capable of determining the corresponding height and alas the cylinder's surface area. (circular "top and bottom" as well as lateral periphery).