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 cm^3 . 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 cm^3
54 cm^3
18 cm^3
24 cm^3
We can use the relationship between the volume of a cylinder and a sphere to solve this problem.
The volume of a cylinder is given by the formula V_cylinder = πr^2h, where r is the radius and h is the height.
The volume of a sphere is given by the formula V_sphere = (4/3)πr^3, where r is the radius.
In this problem, we are given that the volume of the cylinder is 36 cm^3. Let's denote the radius of the cylinder and sphere as r, and the height of the cylinder as 2r (since it is equal to the diameter of the sphere).
We can substitute these values into the formula for the volume of a cylinder:
36 cm^3 = πr^2 * 2r
Simplifying this equation, we have:
36 cm^3 = 2πr^3
Now we can solve for r:
18 cm^3 = πr^3
r^3 = 18/π
r ≈ 2.443 cm
Finally, we can substitute this value of r into the formula for the volume of a sphere to find the volume of the sphere:
V_sphere = (4/3)π(2.443)^3
V_sphere ≈ 54 cm^3
Therefore, the volume of the sphere is approximately 54 cm^3.
The correct answer is 54 cm^3.