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 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) 54c * m ^ 3; 24c * m ^ 3; 18c * m ^ 3; 12c * m ^ 3
The volume of a cylinder is given by the formula V_cylinder = π*r^2*h, where r is the radius and h is the height.
In this problem, we are given that the volume of the cylinder is 36c*m^3, so we can write:
36c*m^3 = π*r^2*h
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 write:
h = 2r
Substituting this into the equation for the volume of the cylinder, we have:
36c*m^3 = π*r^2*(2r)
Simplifying this equation, we get:
36c*m^3 = 2π*r^3
Dividing both sides of the equation by 2π, we get:
18c*m^3/π = r^3
Now, we can use the formula for the volume of a sphere, V_sphere = (4/3)π*r^3, to find the volume of the sphere. Substituting the value of r^3 from the previous equation, we have:
V_sphere = (4/3)π*(18c*m^3/π)
Simplifying this equation, we get:
V_sphere = (4/3)*18c*m^3
V_sphere = 72c*m^3/3
V_sphere = 24c*m^3
Therefore, the volume of the sphere is 24c*m^3.