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?
To find the volume of the cone, we first need to find the radius and height of the cone.
Given that the volume of the sphere is 72 m³, we can use the formula for the volume of a sphere: V = (4/3)πr³. Solving for r, we get:
72 = (4/3)πr³
r³ = (3/4)(72/π)
r = ∛(27/π)
r ≈ 2.084 meters
Since the radius of the cone is equal to the radius of the sphere, the radius of the cone is also approximately 2.084 meters.
Given that the height of the cone is equal to the diameter of the sphere, we know that the diameter of the sphere is twice the radius, so the height of the cone is twice the radius. Therefore, the height of the cone is approximately 2 * 2.084 = 4.168 meters.
Now, we can calculate the volume of the cone using the formula for the volume of a cone: V = (1/3)πr²h. Plugging in the values for r and h, we get:
V = (1/3)π(2.084)²(4.168)
V ≈ 23 m³
Therefore, the volume of the cone is approximately 23 m³.