The volume of a sphere is 48 . What is the volume of a cylinder if its radius is the same as the sphere's and its height is equal to the sphere's diameter ? ( 1 point )
To find the volume of a cylinder, we need to know the formula for the volume of a cylinder. The formula for the volume of a cylinder is V = πr²h, where V is the volume, r is the radius, and h is the height.
In this case, the radius of the cylinder is the same as the sphere's radius and the height of the cylinder is equal to the sphere's diameter. Let's call the radius of the sphere (and cylinder) as "r".
We are given that the volume of the sphere is 48. Using the formula for the volume of a sphere, we can write it as:
V_sphere = (4/3) * π * r³ = 48
Now, let's find the volume of the cylinder. The radius of the cylinder is also "r" and the height of the cylinder is equal to the diameter of the sphere, which is 2r. Let's substitute these values into the formula for the volume of a cylinder:
V_cylinder = π * r² * 2r
V_cylinder = 2π * r³
Since we know that the volume of the sphere is 48, we can substitute this value into the equation:
48 = 2π * r³
Now, we can solve for r:
48 / (2π) = r³
24 / π = r³
Now, let's plug this value of r into the equation for the volume of the cylinder:
V_cylinder = 2π * (24 / π)³
V_cylinder = 2π * (24/π)^3
V_cylinder = 2 * 24^3 * π / π^2
V_cylinder = 2 * (13824π) / π^2
V_cylinder = 27648π / π^2
This is the volume of the cylinder in terms of π.