Question

The relationships of the volumes of cylinders, cones, and spheres (considering that the height and radius are the same for all of the objects):

Put the items in order from least amount needed to greatest amount needed.

Number of cones needed to equal the volume of one sphere
Number of cones needed to equal the volume of one cylinder
Number of cones needed, along with one sphere, to equal the volume of one cylinder

Answer 1

Take the common radius r and set the cone and cylinder height equal to the sphere's diameter, h = 2r (so all three share the same r and h).

Volumes:
- Cone: Vcone = (1/3)πr^2h = (1/3)πr^2(2r) = (2/3)πr^3
- Sphere: Vsphere = (4/3)πr^3
- Cylinder: Vcyl = πr^2h = πr^2(2r) = 2πr^3

Numbers:
- Cones to equal one sphere: Vsphere / Vcone = (4/3)/(2/3) = 2
- Cones to equal one cylinder: Vcyl / Vcone = 2 / (2/3) = 3
- Cones needed along with one sphere to equal one cylinder: solve n·(2/3) + 4/3 = 2 → n = 1

Order from least amount needed to greatest:
1) Number of cones needed, along with one sphere, to equal one cylinder (1)
2) Number of cones needed to equal one sphere (2)
3) Number of cones needed to equal one cylinder (3)