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?(1 point)
Responses
A. 24 cm3
B. 18 cm3
C. 54 cm3
D. 12 cm3
C. 54 cm3
Explanation: Since the height of the cylinder is equal to the diameter of the sphere, the diameter of the sphere is equal to the radius of the cylinder. Therefore, the radius of the sphere is the same as the radius of the cylinder.
The formula for the volume of a cylinder is V_cylinder = πr^2h, where r is the radius and h is the height. Given that V_cylinder = 36 cm^3, we can solve for the radius r by rearranging the formula:
36 = πr^2(2r)
36 = 2πr^3
r^3 = 36 / (2π)
r = (36 / (2π))^(1/3)
r ≈ 2.14 cm
Now that we know the radius of the sphere is approximately 2.14 cm, we can calculate the volume of the sphere using the formula V_sphere = (4/3)πr^3:
V_sphere = (4/3)π(2.14)^3
V_sphere ≈ 54 cm^3
Therefore, the volume of the sphere is 54 cm^3.