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?
The relationship between the volume formulas for the sphere, cylinder, and cone is as follows:
The volume of the sphere is given by the formula:
V_sphere = (4/3)πr^3
The volume of the cylinder is given by the formula:
V_cylinder = πr^2h
where h is the height of the cylinder.
The volume of the cone is given by the formula:
V_cone = (1/3)πr^2h
where h is the height of the cone.
In this case, the height of both the cylinder and the cone is twice the radius. Therefore, using the same radius (r) and height (h) values for both the cylinder and the cone, we can simplify the volume formulas as follows:
V_cylinder = πr^2 * 2r = 2πr^3
V_cone = (1/3)πr^2 * 2r = (2/3)πr^3
Now comparing the volume formulas, we have:
V_sphere : V_cylinder : V_cone = (4/3)πr^3 : 2πr^3 : (2/3)πr^3
Simplifying further, we get:
V_sphere : V_cylinder : V_cone = (4/3) : 2 : (2/3)
Therefore, the relationship between the volume formulas for the sphere, cylinder, and cone is:
V_sphere : V_cylinder : V_cone = (4/3) : 2 : (2/3)
From this relationship, we can see that the volume of the cylinder is twice that of the sphere, while the volume of the cone is (2/3) times that of the sphere.