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
24 cm3
24 cm cubed
54 cm3
54 cm cubed
12 cm3
12 cm cubed
18 cm3
To solve this problem, we need to use the relationship between the volumes of a cylinder and a sphere. The formula for the volume of a cylinder is V = πr^2h, where r is the radius and h is the height. The formula for the volume of a sphere is V = (4/3)πr^3, where r is the radius.
Given that the volume of the cylinder is 36 cm^3, we know that V = 36 cm^3. We also know that the height of the cylinder is equal to the diameter of the sphere, which means that h = 2r.
Substituting these values into the formula for the volume of the cylinder, we have V = πr^2(2r) = 36 cm^3.
Simplifying the equation, we get 2πr^3 = 36. Dividing both sides by 2π, we have r^3 = 18. Taking the cube root of both sides, we find r = ∛18, which is approximately 2.62 cm.
Now that we have the radius of the sphere, we can substitute it into the formula for the volume of a sphere to find the answer. V = (4/3)π(2.62)^3 ≈ 24.16 cm^3.
Therefore, the volume of the sphere is approximately 24 cm^3.