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. 12 cm3
c. 54 cm3
d. 18 cm3
To solve this problem, we need to use the relationships among the volume formulas for cones, cylinders, and spheres.
The volume of a cylinder is given by the formula V = πr^2h, where r is the radius and h is the height.
Since the height of the cylinder is equal to the diameter of the sphere (which is twice the radius), the height of the cylinder is 2r.
Given that the volume of the cylinder is 36 cm^3, we can plug in the values and solve for r:
36 = πr^2(2r)
36 = 2πr^3
r^3 = 18/π
r = (18/π)^(1/3)
Now that we have found the radius, we can use the volume formula for a sphere, which is V = (4/3)πr^3. Plugging in the radius we found, we get:
V = (4/3)π((18/π)^(1/3))^3
V = (4/3)π(18/π)
V = 24 cm^3
Therefore, the volume of the sphere is 24 cm^3, which corresponds to option a. 24 cm^3.