using 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 sm^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
To solve this problem, we need to use the relationship between the volumes of a cylinder, cone, and sphere:
1. Volume of a cylinder: V_cylinder = π * r^2 * h
2. Volume of a cone: V_cone = (1/3) * π * r^2 * h
3. Volume of a sphere: V_sphere = (4/3) * π * r^3
Given that the volume of the cylinder is 36 cm^3, we can substitute the values into the cylinder volume formula:
36 = π * r^2 * h
We then know that the height of the cylinder is equal to the sphere's diameter, so h = 2r.
Substituting this into the equation, we have:
36 = π * r^2 * 2r
Simplifying further:
36 = 2π * r^3
Dividing both sides of the equation by 2π:
r^3 = 18/π
Now we can find the volume of the sphere using the sphere volume formula:
V_sphere = (4/3) * π * r^3
Substituting the value of r^3 we found:
V_sphere = (4/3) * π * (18/π)
Simplifying:
V_sphere = 4 * r^3
Therefore, the volume of the sphere is 4 * (18/π) cm^3.