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: cylinder = sphere - cone
B: sphere = cylinder + cone
C: cone = sphere - cylinder
D: cone = cylinder - sphere
The correct option is:
D: cone = cylinder - sphere
The formula for the volume of a cylinder is V = πr^2h, the formula for the volume of a cone is V = (1/3)πr^2h, and the formula for the volume of a sphere is V = (4/3)πr^3.
Given that the radius (r) is the same for all three shapes and the height (h) of the cylinder and cone is twice the radius, we can calculate the volume for all three shapes:
- Volume of cylinder = πr^2 * 2r = 2πr^3
- Volume of cone = (1/3) * πr^2 * 2r = (2/3) * πr^3
- Volume of sphere = (4/3) * πr^3
Therefore, cone = cylinder - sphere:
(2/3) * πr^3 = 2πr^3 - (4/3) * πr^3
(2/3) * πr^3 = (6/3) * πr^3 - (4/3) * πr^3
(2/3) * πr^3 = (2/3) * πr^3
Hence, cone = cylinder - sphere.