Which option most correctly describes the relationship between the formulas for the volumes of cylinders to that of spheres and cones, assuming the diameter of the sphere is equal to the height of both the cylinder and the cone
The formula for the volume of a cylinder is Vcylinder = πr^2h, where r is the radius of the base and h is the height.
The formula for the volume of a sphere is Vsphere = (4/3)πr^3, where r is the radius.
The formula for the volume of a cone is Vcone = (1/3)πr^2h, where r is the radius of the base and h is the height.
If the diameter of the sphere is equal to the height of both the cylinder and the cone, that means the radius of the sphere (r) is half the height.
If we substitute this information into the formulas:
For the cylinder:
Vcylinder = π(0.5h)^2h = (1/4)πh^3
For the sphere:
Vsphere = (4/3)π(0.5h)^3 = (1/6)πh^3
For the cone:
Vcone = (1/3)π(0.5h)^2h = (1/12)πh^3
In this case, we can see that the volume of the sphere (Vsphere) is equal to half the volume of the cylinder (Vcylinder) and is equal to a third of the volume of the cone (Vcone). Therefore, the relationship between the formulas for the volumes of cylinders to that of spheres and cones, assuming the diameter of the sphere is equal to the height of both the cylinder and the cone, is that the volume of the sphere is half the volume of the cylinder and a third of the volume of the cone.