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
18 cm3
18 cm cubed
54 cm3
54 cm cubed
24 cm3
24 cm cubed
12 cm3
The formula for the volume of a cylinder is V = πr^2h, where r is the radius and h is the height. Given the volume of the cylinder is 36 cm^3, we can solve for the height of the cylinder using the formula:
36 = πr^2h
Since the height of the cylinder is equal to the sphere's diameter, we can use the formula for the volume of a sphere to find its volume. The formula for the volume of a sphere is V = (4/3)πr^3.
The radius of the sphere is the same as the radius of the cylinder, so we can substitute the value of h from the cylinder formula into the sphere formula as the diameter is equal to the height.
V_sphere = (4/3)πr^3 = (4/3)πr^3 = (4/3)π(36/π)^(3/2) = 4/3 * 36^(3/2) cm^3
Therefore, the volume of the sphere is 54 cm^3.