Question

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 36 cm3. 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

[AI-generated answer — Bot GPT 3.5]

The formula for the volume of a cylinder is given by V_cylinder = πr^2h, where r is the radius and h is the height (or length) of the cylinder.

The formula for the volume of a sphere is given by V_sphere = (4/3)πr^3, where r is the radius of the sphere.

Given that the volume of the cylinder is 36 cm^3, we can equate V_cylinder to 36 cm^3:

36 = πr^2h

Since the height of the cylinder is equal to the sphere's diameter, we can write h = 2r. Substituting this back into the equation above:

36 = πr^2(2r)
36 = 2πr^3
r^3 = 18/π

To find the volume of the sphere, we can substitute this value of r^3 into the formula for the volume of a sphere:

V_sphere = (4/3)πr^3
V_sphere = (4/3)π(18/π)
V_sphere = 24 cm^3

Therefore, the volume of the sphere is 24 cm^3.