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 12 cm3 12 cm cubed 24 cm3 24 cm cubed 18 cm3 18 cm cubed 54 cm3
First, we need to find the radius and height of the cylinder using its volume formula: V_cylinder = πr^2h. Given that the volume of the cylinder is 36 cm^3, we have:
36 = πr^2h
Since the height of the cylinder is equal to the sphere's diameter, the height can be expressed as 2r. Therefore, we can rewrite the equation as:
36 = πr^2(2r)
36 = 2πr^3
r^3 = 18/π
r = (18/π)^(1/3)
Now that we have found the radius of the cylinder, we can use the formula for the volume of a sphere (V = (4/3)πr^3) to find the volume of the sphere with the same radius:
V_sphere = (4/3)π((18/π)^(1/3))^3
V_sphere = (4/3)π(18/π)
V_sphere = 24 cm^3
Therefore, the volume of the sphere is 24 cm^3.