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
18 cm3
18 cm cubed
24 cm3
24 cm cubed
54 cm3
54 cm cubed
12 cm3
To solve this problem, we need to use the relationship between the volumes of a cylinder and a sphere.
The volume of a cylinder is given by the formula Vcylinder = πr^2h, where r is the radius and h is the height of the cylinder. We are given that the volume of the cylinder is 36 cm^3.
The volume of a sphere is given by the formula Vsphere = (4/3)πr^3, where r is the radius of the sphere.
We are also given that the height of the cylinder is equal to the diameter of the sphere. Since the diameter of a sphere is twice the radius, we can say that h = 2r.
We can substitute the value of h from the cylinder into the formula for the sphere's volume to get:
Vsphere = (4/3)πr^3
Vsphere = (4/3)π(r^2 * 2r)
Vsphere = (4/3)π(2r * r^2)
Vsphere = (8/3)πr^3
Now, let's equate the volume of the cylinder with the volume of the sphere:
36 = (8/3)πr^3
To find the volume of the sphere, we can solve for r:
r^3 = (3/8) * (36/π)
r^3 = 9
r = 3
Now, we can substitute the value of r back into the formula for the sphere's volume:
Vsphere = (8/3)π(3)^3
Vsphere = (8/3)π(27)
Vsphere = 72π
Therefore, the volume of the sphere is 72π cm^3.