The volume of a sphere is 72 m3 . What is the volume of a cone if its radius is the same as the sphere’s and its height is equal to the sphere’s diameter
The volume of a sphere is given by the formula: V = (4/3)πr³, where V is the volume and r is the radius.
Let's find the radius of the sphere first. We can rearrange the formula to solve for r:
V = (4/3)πr³
72 = (4/3)πr³
r³ = (3/4) * 72 / π
r³ = 54 / π
r = (54 / π)^(1/3)
Now, let's find the diameter of the sphere:
Diameter = 2 * radius
Diameter = 2 * (54 / π)^(1/3)
The height of the cone is equal to the sphere's diameter, so the height of the cone is:
Height = Diameter = 2 * (54 / π)^(1/3)
Now, let's find the formula for the volume of a cone:
V_cone = (1/3) * π * r² * h
Substituting the values:
V_cone = (1/3) * π * (54 / π)^(1/3)² * 2 * (54 / π)^(1/3)
Simplifying:
V_cone = (1/3) * π * (54 / π)^(2/3) * 2 * (54 / π)^(1/3)
V_cone = (2/3) * π * (54 / π) * (54 / π)^(2/3)
V_cone = (2/3) * (54 / π) * (54 / π)^(2/3)
V_cone = (2/3) * 54 * (54 / π)^(2/3)
Therefore, the volume of the cone is (2/3) * 54 * (54 / π)^(2/3) cubic units.