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
The volume of a sphere is given by the formula V_sphere = (4/3)πr^3, where r is the radius.
The volume of a cylinder is given by the formula V_cylinder = πr^2h, where r is the radius and h is the height.
The volume of a cone is given by the formula V_cone = (1/3)πr^2h, where r is the radius and h is the height.
In this scenario, the radius is the same for all three shapes, so we can compare the volume formulas by substituting the value of the given height (which is twice the radius) into the volume formulas.
Substituting the value of h = 2r into the volume formulas:
V_sphere = (4/3)πr^3
V_cylinder = πr^2(2r) = 2πr^3
V_cone = (1/3)πr^2(2r) = (2/3)πr^3
So, the relationship between the volume formulas for the sphere, cylinder, and cone is:
V_sphere : V_cylinder : V_cone = 1 : 2 : 2/3