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 it's radius is the same as the cylinder's and the height of the cylinder is equal to the spheres diameter?
The relationship among the volumes of cones, cylinders, and spheres with the same radius (r) and the height of the cylinder (h) equal to the diameter of the sphere (2r) states that:
Volume of a cylinder (V_cylinder) = πr²h
Volume of a sphere (V_sphere) = (4/3)πr³
According to the problem, the height of the cylinder is equal to the diameter of the sphere, which means h = 2r. Therefore, we can substitute h in the volume formula for the cylinder:
V_cylinder = πr²(2r)
Given that the volume of the cylinder is 36 cm³, we have:
36 = πr²(2r) = 2πr³
Divide both sides of the equation by 2π to solve for r³:
r³ = 36 / (2π)
Now we can calculate the volume of the sphere using the value of r³:
V_sphere = (4/3)πr³
Substitute the value of r³ that we computed:
V_sphere = (4/3)π * (36 / (2π))
Simplify by canceling π and simplifying the constants:
V_sphere = (4/3) * (36 / 2)
V_sphere = (4/3) * 18
V_sphere = (4 * 18) / 3
V_sphere = 72 / 3
V_sphere = 24 cm³
The volume of the sphere with the same radius as the cylinder and with its diameter equal to the height of the cylinder is 24 cm³.