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 18 cm3 18 cm cubed 54 cm3 54 cm cubed 12 cm3
First, we need to find the radius and height of the cylinder. Since the volume of the cylinder is given by V = πr^2h, and we know that V = 36 cm^3, we can rearrange the formula to solve for h: h = V / (πr^2) = 36 / (πr^2).
Since the height of the cylinder is equal to the diameter of the sphere, and the diameter is twice the radius, we have h = 2r. Therefore, we can write 2r = 36 / (πr^2) and solve for r:
2r * πr^2 = 36
2πr^3 = 36
r^3 = 36 / 2π
r^3 = 18 / π
r = (18 / π)^(1/3)
r ≈ 1.751 cm
Now that we have found the radius of the cylinder, we can use the formula for the volume of a sphere, which is V = (4/3)πr^3. Plugging in the radius we found, we get:
V = (4/3)π(1.751)^3
V ≈ 22.76 cm^3
Therefore, the volume of the sphere is approximately 22.76 cm^3.