Use the relationship among the formulas for the volumes of cones, cylinders, and spheres to solve the following problem. The volume of a cylinder is 36 cm3 . What is the volume of a sphere if its radius is the same as the cylinder’s and the height of the cylinder is equal to the sphere’s diameter?(1 point) Responses 12 cm3 12 cm cubed 24 cm3 24 cm cubed 54 cm3 54 cm cubed 18 cm3 18 cm cubed
The formula for the volume of a cylinder is Vcylinder = πr²h, where r is the radius and h is the height.
The formula for the volume of a sphere is Vsphere = 4/3πr³, where r is the radius.
In this problem, given that the volume of the cylinder is 36 cm³, it means that Vcylinder = 36 cm³.
It is also given that the height of the cylinder is equal to the diameter of the sphere, which means h = 2r.
Substituting these values into the formula for the volume of the cylinder, we get:
36 = πr²(2r)
36 = 2πr³
18 = πr³
Solving for r,
r³ = 18/π
r = (18/π)^(1/3)
Now, substituting this value of r into the formula for the volume of the sphere, we get:
Vsphere = 4/3πr³
Vsphere = 4/3π((18/π)^(1/3))³
Vsphere = 4/3π(18/π)
Vsphere = 4/3 * 18
Vsphere = 72/3
Vsphere = 24 cm³
Therefore, the volume of the sphere is 24 cm cubed.