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
24 cm3
24 cm cubed
18 cm3
To solve this problem, we need to use the relationship between the volumes of a cylinder and a sphere.
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.
Given that the volume of the cylinder is 36 cm^3 and the height of the cylinder is equal to the sphere's diameter, we can find the radius of the cylinder (and sphere) by using the formula for the volume of the cylinder.
36 = πr^2h
Since the height of the cylinder is equal to the sphere's diameter, we can substitute 2r for h.
36 = πr^2(2r)
36 = 2πr^3
r^3 = 18/π
r = (18/π)^(1/3)
Now that we have the radius of the sphere, we can substitute this value into the formula for the volume of a sphere to find the volume of the sphere.
Vsphere = (4/3)πr^3
Vsphere = (4/3)π((18/π)^(1/3))^3
Vsphere = (4/3)π(18/π)
Vsphere = (4/3)(18)
Vsphere = 72/3
Vsphere = 24 cm^3
Therefore, the volume of the sphere is 24 cm^3.