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. Sphere = cylinder + cone
D. Cylinder = sphere - cone
To compare the volumes of the three shapes, we need to consider their volume formulas.
The volume of a sphere is given by the formula V = (4/3)πr^3, where r is the radius.
The volume of a cylinder is given by the formula V = πr^2h, where r is the radius and h is the height.
The volume of a cone is given by the formula V = (1/3)πr^2h, where r is the radius and h is the height.
Since the cylinder and cone have the same height (h = 2r), we can substitute this value into their volume formulas:
Volume of cylinder = πr^2(2r) = 2πr^3
Volume of cone = (1/3)πr^2(2r) = (2/3)πr^3
Comparing the formulas, we can see that the volume of the cone is (2/3) of the volume of the cylinder.
Therefore, the correct relationship between the volume formulas is:
Cylinder = sphere + cone
Therefore, the correct answer is C.