Suppose a sphere, cylinder, and cone all share the same radius and 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?(1 point) Responses: a:3 cones = 1 cylinder b:1 cylinder = 2 cones + 1 sphere c:sphere = cylinder + cone d:sphere - cone = cylinder
The answer is a: 3 cones = 1 cylinder.
The volume formulas for the sphere, cylinder, and cone are as follows:
Sphere: V = (4/3)πr^3
Cylinder: V = πr^2h
Cone: V = (1/3)πr^2h
Given that the radius is the same for all three shapes, we can compare their volumes.
Since the height of the cylinder and cone is twice the radius, we can substitute 2r for h in the volume formulas.
For the sphere: V_sphere = (4/3)πr^3
For the cylinder: V_cylinder = πr^2(2r) = 2πr^3
For the cone: V_cone = (1/3)πr^2(2r) = (2/3)πr^3
Dividing the volume of the cylinder by the volume of the cone, we get:
V_cylinder / V_cone = (2πr^3) / ((2/3)πr^3) = 3
Therefore, 3 cones are equal to 1 cylinder.