Suppose a sphere and a cone share the same radius and the height of the cone is twice the radius. Which ratio describes the relationship between the formula's for the volume of a cone to the volume of a sphere?
The volume of a sphere is given by the formula V_sphere = (4/3)πr³, where r is the radius of the sphere.
The volume of a cone is given by the formula V_cone = (1/3)πr²h, where r is the radius of the base of the cone and h is the height of the cone.
In this case, the radius of both the sphere and the cone is the same.
Given that the height of the cone is twice the radius, we have h = 2r.
Substituting this value of h into the formula for the volume of the cone, we get:
V_cone = (1/3)πr²(2r) = (2/3)πr³
Comparing this to the formula for the volume of the sphere, we can conclude:
V_cone : V_sphere = (2/3)πr³ : (4/3)πr³
Simplifying the expression, we have:
V_cone : V_sphere = 2 : 4
Therefore, the ratio that describes the relationship between the formulas for the volume of a cone and the volume of a sphere is 2 : 4, which can be simplified to 1 : 2.