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
54 cm3
54 cm cubed
12 cm3
12 cm cubed
18 cm3
18 cm cubed
24 cm3
To solve this problem, we need to understand the relationship between the formulas for the volume of a cylinder, cone, and sphere.
The volume of a cylinder is given by the formula: Vcylinder = πr²h, where r is the radius and h is the height.
The volume of a cone is given by the formula: Vcone = 1/3πr²h, where r is the radius and h is the height.
The volume of a sphere is given by the formula: Vsphere = 4/3πr³, where r is the radius.
In this problem, we are given the volume of the cylinder as 36 cm³. We are also told that the height of the cylinder is equal to the sphere's diameter, which means h = 2r (since the diameter is twice the radius).
To find the volume of the sphere, we need to find the radius first. We can rearrange the formula for the volume of a cylinder to solve for the radius:
Vcylinder = πr²h
36 = πr²(2r)
36 = 2πr³
r³ = 36/(2π)
r³ = 18/π
r = (18/π)^(1/3)
Now, substituting this value of r into the formula for the volume of a sphere:
Vsphere = 4/3πr³
Vsphere = 4/3π((18/π)^(1/3))³
Vsphere = 4/3π(18/π)
Vsphere = 24 cm³
Therefore, the volume of the sphere is 24 cm³.