a cylinder and a cone start with the same radius and height the radius of the cone is then trippled and the height of the cone is cut in half. the radius of the cylinder stays the same

Therefore the cone is the greatest relative increase in volume.

Step-by-step explanation:

Cone:

Original cone = (1/3) π (h) r^2

Changed cone = (1/3) π (h/2) (3r) ^2

= (1/2) (1/3) π (h) 9r^2

= (9/2) * Original cone

=4.5 * Original cone

Cylinder:

Original cylinder = π (h) r^2

Changed cylinder = π (2h) r^2

=2 * Original cylinder

Therefore the cone is the greatest relative increase in volume.