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 54 cm3 54 cm cubed 24 cm3 24 cm cubed 18 cm3 18 cm cubed 12 cm3
To solve this problem, we know that the volume of a cylinder is given by the formula V = πr^2h, where r is the radius and h is the height.
Given that the volume of the cylinder is 36 cm^3 and the height of the cylinder is equal to the sphere's diameter (which is twice the sphere's radius), we can write the following equation: 36 = πr^2(2r).
Simplifying this equation, we get: 36 = 2πr^3.
Solving for r, we find that r = ∛(18/π) = ∛(18) / ∛π.
Now, to find the volume of a sphere with the same radius, we use the formula for the volume of a sphere: V = (4/3)πr^3.
Substitute the value of r we found earlier, we get: V = (4/3)π(∛(18))^3 = (4/3)π(18) = 24π ≈ 75.4 cm^3.
Therefore, the volume of the sphere with the same radius as the cylinder and the height equal to the sphere's diameter is approximately 75.4 cm^3.