Suppose a sphere, cylinder and cone all share the same radius and 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?
1. 3 cones = 1 cylinder
2. 1 cylinder - 2 cone + 1 sphere
3. sphere - cone = cone
4. Sphere = cylinder + cone
The correct answer is 4. Sphere = cylinder + cone.
The volume formulas for the sphere, cylinder, and cone are given by:
Sphere: V = (4/3)πr^3
Cylinder: V = πr^2h
Cone: V = (1/3)πr^2h
In this case, the radius of the sphere, cylinder, and cone is the same, let's call it r.
The height of the cylinder and cone is given as twice the radius, so h = 2r.
Substituting these values into the volume formulas, we get:
Sphere: V = (4/3)πr^3
Cylinder: V = πr^2(2r) = 2πr^3
Cone: V = (1/3)πr^2(2r) = (2/3)πr^3
Adding the volume of the cylinder and cone, we have:
Cylinder + Cone = 2πr^3 + (2/3)πr^3 = (8/3)πr^3
Comparing this with the volume of the sphere, we have:
Sphere = (4/3)πr^3
Therefore, Sphere = Cylinder + Cone.