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?(1 point)
Responses
First, we need to find the radius of the cylinder. The formula for the volume of a cylinder is V = πr^2h, where V is the volume, r is the radius, and h is the height. Since the volume of the cylinder is 36 cm^3 and the height is equal to the sphere's diameter, we can solve for the radius:
36 = πr^2(h)
36 = πr^2(2r)
36 = 2πr^3
r^3 = 18/π
r = (18/π)^(1/3)
Now, since the radius of the sphere is the same as the cylinder's, the radius of the sphere is also (18/π)^(1/3). The formula for the volume of a sphere is V = (4/3)πr^3, where V is the volume and r is the radius. Substituting in the radius of the sphere, we can find the volume:
V = (4/3)π(18/π)^(1/3)^3
V = (4/3)π(18/π)
V = 24 cm^3
Therefore, the volume of the sphere is 24 cm^3.