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 formulas for the volume of a cone to the volume of a sphere
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The volume of a sphere is given by the formula:
V_sphere = (4/3) * π * r^3
where r is the radius of the sphere.
The volume of a cone is given by the formula:
V_cone = (1/3) * π * r^2 * h
where r is the radius of the base of the cone and h is the height of the cone.
In this case, it is mentioned that the height of the cone is twice the radius, which means h = 2r.
Now, let's calculate the ratio between the formulas for the volume of a cone to the volume of a sphere:
V_cone / V_sphere = [ (1/3) * π * r^2 * h ] / [ (4/3) * π * r^3 ]
Canceling out common terms:
V_cone / V_sphere = (1/3) * r^2 * h / r^3
Substituting h = 2r:
V_cone / V_sphere = (1/3) * r^2 * (2r) / r^3
Simplifying:
V_cone / V_sphere = (1/3) * 2r^3 / r^3
V_cone / V_sphere = (1/3) * 2
V_cone / V_sphere = 2/3
Therefore, the ratio between the formulas for the volume of a cone to the volume of a sphere is 2/3.