The volume of a sphere is 72 m". 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? (1 point)
Let's solve this step by step.
Given:
- The volume of the sphere is 72 m³.
- The radius of the sphere is r.
The volume of a sphere is given by the formula: V = (4/3)πr³.
So, we can set up the equation 72 = (4/3)πr³.
Now, we need to find the radius of the sphere.
72 = (4/3)πr³
72 * 3/4π = r³
54π = r³
r = ³√(54π)
Now, let's move on to the volume of the cone.
The volume of a cone is given by the formula: V = (1/3)πr²h, where r is the radius and h is the height.
In this case, the radius of the cone is the same as the sphere's radius, which is ³√(54π), and the height is equal to the sphere's diameter, which is 2 times the radius, so the height is 2 * ³√(54π).
Therefore, the volume of the cone is given by:
V = (1/3)π * (³√(54π))² * (2 * ³√(54π))
V = (1/3)π * (³√(54π))² * 2 * ³√(54π)
V = (1/3)π * (³√(54π))² * 6 * √(6π)
V = (1/3)π * (³√54)² * 6 * √(π * π * 6)
V = (1/3)π * 9 * 6 * √(π * π * 6)
V = 54π√(36π)
So, the volume of the cone is 54π√(36π) cubic units.