Use a relationship among the formulas for the volumes of cone cylinders and spheres to solve the following problem the volume of the cylinder is 36 cm cubed what is the volume of the sphere of the radius is the same as the cylinders and the height of the cylinders equal to the spheres diameter
A. 18 cm cubed
B. 54cm cubed
C. 12 cm cubed
D. 24 cm cubed
First, let's establish the relationship between the formulas for the volumes of a cone, cylinder, and sphere:
The volume of a cone is given by the formula V_cone = (1/3)πr^2h, where r is the radius of the base and h is the height.
The volume of a cylinder is given by the formula V_cylinder = πr^2h.
The volume of a sphere is given by the formula V_sphere = (4/3)πr^3, where r is the radius of the sphere.
Now, let's use the given information to determine the volume of the sphere. We are given that the volume of the cylinder is 36 cm^3, which means V_cylinder = 36.
Since the height of the cylinder is equal to the diameter of the sphere, we can say that h = 2r.
Substituting this relationship into the formula for the volume of a cylinder, we get V_cylinder = πr^2(2r) = 2πr^3.
Since V_cylinder = 36, we have 2πr^3 = 36.
Dividing both sides of the equation by 2π, we get r^3 = 18/π.
Taking the cube root of both sides of the equation, we get r = (18/π)^(1/3).
Now, substituting this value of r into the formula for the volume of a sphere, we get V_sphere = (4/3)π((18/π)^(1/3))^3.
Simplifying this expression, we get V_sphere = (4/3)π(18/π) = 72/3 = 24.
Therefore, the volume of the sphere is 24 cm^3.
The correct answer is D. 24 cm^3.