Suppose a sphere, cylinder, and cone all share the same radius. The cylinder and cone also share the same height, which is twice the radius. Which of the following is true about the relationship between the volume formulas for the sphere, cylinder, and cone?
a. Cone = Cylinder - Sphere
b. Cone = Sphere - Cylinder
c. Cylinder = Sphere - Cone
d. Sphere = cylinder + Cone
The volume formula for a sphere is V = (4/3)πr^3.
The volume formula for a cylinder is V = πr^2h.
The volume formula for a cone is V = (1/3)πr^2h.
Given that the radius is the same for all three shapes and the height of the cylinder and cone is twice the radius, we can substitute 2r for h in the formulas for the cylinder and cone.
Now, the volume formula for the cylinder becomes V = πr^2(2r) = 2πr^3.
And the volume formula for the cone becomes V = (1/3)πr^2(2r) = (2/3)πr^3.
From these formulas, we can see that the volume of the cylinder is twice the volume of the cone, and the volume of the cone is (2/3) times the volume of the sphere.
Therefore, the correct relationship between the volume formulas is:
c. Cylinder = Sphere - Cone