the volume of a sphere is 48 m^3. 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
The volume of a sphere is given by the formula V = 4/3 * π * r^3, where V is the volume, π is a mathematical constant approximately equal to 3.14159, and r is the radius of the sphere.
We are given that the volume of the sphere is 48 m^3. So, we can set up the equation:
48 = 4/3 * π * r^3
To find the volume of the cylinder, we first need to find the radius of the sphere. Rearranging the equation above, we can solve for r:
48 = 4/3 * π * r^3
48 / (4/3 * π) = r^3
(48 * 3) / (4 π) = r^3
36 / π = r^3
r = (36 / π)^(1/3)
Now, we need to find the height of the cylinder. The height is equal to the sphere's diameter, and since the diameter is twice the radius, the height of the cylinder is 2 * (36 / π)^(1/3).
The volume of a cylinder is given by the formula V = π * r^2 * h, where V is the volume, r is the radius, and h is the height.
So, the volume of the cylinder with the same radius as the sphere and a height equal to the sphere's diameter is:
V = π * (36 / π)^(1/3)^2 * 2 * (36 / π)^(1/3)
Simplifying the expression:
V = π * (36 / π)^(2/3) * 2 * (36 / π)^(1/3)
V = 2 * π * (36 / π)^((2/3) + (1/3))
V = 2 * π * (36 / π)^(3/3)
V = 2 * π * (36 / π)^1
V = 2 * π * (36 / π)
V = 2 * 36
V = 72
Therefore, the volume of the cylinder is 72 m^3.