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?
18 cm3
18 cm cubed
12 cm3
12 cm cubed
54 cm3
54 cm cubed
24 cm3
The volume of a cylinder is given by the formula V = πr^2h, where r is the radius and h is the height. Given that the volume of the cylinder is 36 cm^3, we have:
36 = πr^2h
Since the height of the cylinder is equal to the diameter of the sphere, we can write h = 2r. Substituting this into the equation above, we get:
36 = πr^2 * 2r
36 = 2πr^3
r^3 = 18/π
Now, the volume of a sphere is given by the formula V = (4/3)πr^3. Substituting the value of r^3 that we found above, we get:
V = (4/3)π * (18/π)
V = 24
Therefore, the volume of the sphere is 24 cm^3.