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
18 cm3
18 cm cubed
54 cm3
54 cm cubed
24 cm3
The volume of a cylinder is given by the formula V = πr^2h, where r is the radius and h is the height. Given that the volume of the cylinder is 36 cm^3, we can substitute the volume as 36 and since the height is equal to twice the radius, we can say h = 2r.
36 = πr^2(2r)
36 = 2πr^3
r^3 = 18/π
r = (18/π)^(1/3)
The volume of a sphere is given by the formula V = (4/3)πr^3. Given that the radius of the sphere is the same as the cylinder, we can substitute r = (18/π)^(1/3) into the formula for the sphere's volume.
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.