Use the relationship among the formulas for the volumes of cones, cylinders, and spheres to solve the following problem The volume of a s 36c * m ^ 3 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) 24c * m ^ 3 12c * m ^ 3
18c * m ^ 3
54c * m ^ 3
We know that the volume of a cylinder can be calculated using the formula:
Vcylinder = πr^2h
The volume of a cone can be calculated using the formula:
Vcone = (1/3)πr^2h
And the volume of a sphere can be calculated using the formula:
Vsphere = (4/3)πr^3
Given that the radius of the sphere is the same as the cylinder's, and the height of the cylinder is equal to the sphere's diameter, we can use this relationship to solve the problem.
Let's assume the radius of the sphere and cylinder is r, and the height of the cylinder is h.
Since the height of the cylinder is equal to the sphere's diameter, we have:
h = 2r
Substituting this into the volume formula for the cylinder, we get:
Vcylinder = πr^2(2r) = 2πr^3
Since the radius of the sphere and cylinder is the same, we can write the volume of the sphere in terms of r as:
Vsphere = (4/3)πr^3
Comparing the volume formulas for the cylinder and sphere, we can see that the volume of the sphere is (4/3) times the volume of the cylinder.
Therefore, the volume of the sphere is:
Vsphere = (4/3)Vcylinder = (4/3)(2πr^3) = (8/3)πr^3
Given that the volume of the cylinder is 36c * m^3, we can find the volume of the sphere by substituting this value into the expression we derived:
Vsphere = (8/3)πr^3 = (8/3)(36c * m^3) = 96c * m^3
So, the volume of the sphere is 96c * m^3.