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 relationship between the formulas for the volumes of cylinders and spheres, assuming the diameter of the sphere is equal to the height of the cylinder, can be described as follows:
The volume of a cylinder is equal to the product of the cross-sectional area of the circular base (πr^2) and the height (h).
The volume of a sphere is equal to four-thirds of the product of π and the cube of the radius (4/3πr^3).
If the diameter of the sphere is equal to the height of the cylinder, then the radius of the sphere (r) would be half the height of the cylinder (h/2).
Therefore, the formula for the volume of the sphere can be expressed as four-thirds of the product of π and the cube of the radius (4/3π(h/2)^3). Simplifying this gives us the formula for the volume of the sphere as (1/6)πh^3.
The relationship between the formulas for the volumes of the cylinder and the sphere, in this scenario, is that the volume of the sphere is equal to one-sixth of the volume of the cylinder.
As for the relationship between the formulas for the volumes of cylinders and cones, assuming the diameter of the sphere is equal to the height of the cone, the relationship is different. The volume of a cone is equal to one-third of the product of π, the square of the radius (πr^2), and the height (h). If the diameter of the sphere is equal to the height of the cone, then the radius of the sphere (r) would be half the height of the cone (h/2). Therefore, the formula for the volume of the cone can be expressed as one-third of the product of π, the square of the radius, and the height (1/3π(h/2)^2h). Simplifying this gives us the formula for the volume of the cone as (1/12)πh^3.
Therefore, the relationship between the formulas for the volumes of the cylinder and the cone, in this scenario, is that the volume of the cone is equal to one-twelfth of the volume of the cylinder.