Cylinder of maximum volume and maximum lateral area inscribed in a cone

A right circular cylinder of radius r and height h is inscribed in a right circular cone of radius 6 m and height 12 m.

Part 1: Determine the radius of the cylinder such that its volume is a maximum.
A. 2 m
B. 4 m
C. 3 m
D. 5 m

Part 2: Determine the maximum volume of the cylinder.
A. 145.72 m3
B. 321.12 m3
C. 225.31 m3
D. 201.06 m3

Part 3: Determine the height of the cylinder such that its lateral area is a maximum.
A. 10 m
B. 8 m
C. 6 m
D. 4 m