The sphere's radius is one-third the radius of the hemisphere. How does the volume of this hemisphere compare with the volume of the sphere?

Volume of the sphere = 3/4 pi r^3 

The spheres radius is one third of the hemisphere's so radius of hemisphere = 3r 
Volume of the hemisphere = 1/2 * 4/3 pi (3r)^3
                                         =   2/3 * 27 pi r^3
                                         =  18 pi r^3
So vol hemisphere / vol sphere =  18 pi r^3 / 4/3 pi r^3 = 18 * 3/4 =  13.5

Hemisphere is 13.5 times the volume of the sphere.