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?(1 point)
To solve this problem, we need to find the radius and height of the cone.
The volume of a sphere is given by the formula:
V = (4/3) * π * r^3
Where V is the volume and r is the radius.
Since we know the volume of the sphere is 72 m^3, we can set up the equation:
72 = (4/3) * π * r^3
To find the radius, we can divide both sides of the equation by (4/3) * π:
r^3 = (3/4) * 72 / π
r^3 = 54 / π
By taking the cube root of both sides, we can find the value of r:
r = (54 / π)^(1/3)
Now, we know the radius of the cone.
The height of the cone is equal to the diameter of the sphere. Since the diameter is twice the radius, the height of the cone is 2 * r.
Now, we can calculate the volume of the cone.
The volume of a cone is given by the formula:
V = (1/3) * π * r^2 * h
Where V is the volume, r is the radius, and h is the height.
Plugging in the values we have:
V = (1/3) * π * (54 / π)^(1/3)^2 * (2 * (54 / π)^(1/3))
Simplifying the expression, we get:
V = (1/3) * π * (54 / π)^(2/3) * 2 * (54 / π)^(1/3)
V = (2/3) * π * (54 / π)^(2/3 + 1/3)
V = (2/3) * π * (54 / π)^(3/3)
V = (2/3) * π * (54 / π)
V = (2/3) * 54
V = 36
Therefore, the volume of the cone is 36 m^3.