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The formula for the volume of a cylinder is Vcylinder = πr^2h, where r is the radius and h is the height.
The formula for the volume of a sphere is Vsphere = (4/3)πr^3, where r is the radius.
In this problem, the height of the cylinder is equal to the diameter of the sphere, which means the height is twice the radius of the sphere. So, h = 2r.
Given that the volume of the cylinder is 36 cm^3, we can plug this value into the formula for the volume of a cylinder:
36 = πr^2(2r)
36 = 2πr^3
18 = πr^3
To find the volume of the sphere with the same radius, we can substitute the value of r into the formula for the volume of a sphere:
Vsphere = (4/3)πr^3 = (4/3)π(18)
Vsphere = 24π cm^3
So, the volume of the sphere is 24 cm^3.