The volume of a sphere is 48 m°. 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)πr^3, where V is the volume and r is the radius. We are given that the volume of the sphere is 48 m³. So, 48 = (4/3)πr^3.
To find the volume of the cylinder, we need to find its radius and height. It is mentioned that the radius of the cylinder is the same as the sphere's radius. Let's denote this common radius as r_c.
It is also mentioned that the height of the cylinder is equal to the sphere's diameter. The diameter of the sphere is 2 times the radius, so the height of the cylinder would be 2r.
Now, let's find the value of r using the equation 48 = (4/3)πr^3:
48 = (4/3)πr^3
r^3 = (3/4)(48/π)
r^3 = 36/π
r = (36/π)^(1/3)
The radius of the sphere and cylinder are now known, and we know that the height of the cylinder is 2r. Therefore, the height of the cylinder is:
Height = 2r = 2(36/π)^(1/3)
Now, we can use the formula for the volume of a cylinder, V = πr²h:
Volume of the cylinder = π(r_c)^2(Height)
= π(r)^2(2r)
= π[(36/π)^(1/3)]^2(2(36/π)^(1/3))
= π(36/π)^(2/3)(2)(36/π)^(1/3)
= 2π(36/π)^(2/3 + 1/3)
= 2π(36/π)^(3/3)
= 2π(36/π)^1
= 2π(36/π)
= 72
Therefore, the volume of the cylinder is 72 m³.