Use the relashioship among the formulas for the volume of cones cylinders and spheres to solve the following problem the volume of a cylinder is 36cm³ what is the volume of a sphere if it's radius is the same as the cylinders and height of the cylinder is equal to the spheres diameter
To solve this problem, we need to use the relationship between the volume formulas of a cylinder and a sphere.
The volume of a cylinder is given by the formula: V = πr²h,
where V is the volume, r is the radius, and h is the height.
The volume of a sphere is given by the formula: V = (4/3)πr³,
where V is the volume, and r is the radius.
Given that the volume of the cylinder is 36 cm³ and the height of the cylinder is equal to the sphere's diameter, we can determine the radius of the cylinder.
We know that the volume of the cylinder is 36 cm³, so we can equate this to the volume formula of a cylinder:
36 = πr²h
Since the height of the cylinder is equal to the sphere's diameter, we can substitute h with 2r to simplify the equation:
36 = πr²(2r)
36 = 2πr³
r³ = 36/(2π)
r³ = 18/π
r = (18/π)^(1/3)
Now that we have the radius of the cylinder, we can use it to calculate the volume of the sphere. We know that the radius of the sphere is the same as the cylinder, so the radius of the sphere is (18/π)^(1/3).
Using the volume formula of a sphere:
V = (4/3)πr³
Substituting the radius:
V = (4/3)π((18/π)^(1/3))³
Simplifying the expression, we get the volume of the sphere.