Question

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

Answer 1

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.