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?
A. Sphere=cylinder+cone
B. 1 cylinder=2 cones+1 sphere
C. 3 cones = 1 cylinder
D. Sphere - cone = cylinder
The correct answer is C. 3 cones = 1 cylinder.
To see why, let's look at the formulas for the volume of each shape:
- The volume of a sphere with radius r is given by V = (4/3)πr^3.
- The volume of a cylinder with radius r and height h is given by V = πr^2h.
- The volume of a cone with radius r and height h is given by V = (1/3)πr^2h.
Now, if the cylinder and cone share the same height, which is twice the radius, we can substitute 2r for h in the volume formulas for both shapes:
- The volume of the cylinder becomes V = πr^2(2r) = 2πr^3.
- The volume of the cone becomes V = (1/3)πr^2(2r) = (2/3)πr^3.
Notice that the volume of the cylinder is 3 times the volume of the cone. This means that you would need 3 cones to have the same volume as the cylinder. Therefore, the correct relationship between the volume formulas for the sphere, cylinder, and cone is 3 cones = 1 cylinder.