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)%0D%0AResponses%0D%0A%0D%0A12 cm3%0D%0A12 cm cubed%0D%0A%0D%0A18 cm3%0D%0A18 cm cubed%0D%0A%0D%0A54 cm3%0D%0A54 cm cubed%0D%0A%0D%0A24 cm3
The volume of a cylinder is given by the formula V = πr^2h, where r is the radius and h is the height. In this problem, the volume of the cylinder is given as 36 cm^3.
Since the height of the cylinder is equal to the sphere's diameter, we can say that h = 2r.
Substituting this into the formula for the volume of the cylinder, we have 36 = πr^2(2r).
Simplifying this equation, we have 36 = 2πr^3.
Dividing both sides of the equation by 2π, we have 18/π = r^3.
Taking the cube root of both sides of the equation, we have r = (18/π)^(1/3).
Now, we can find the volume of a sphere using the formula V = (4/3)πr^3.
Substituting the value of r from the previous calculation, we have V = (4/3)π((18/π)^(1/3))^3.
Simplifying this equation, we have V = (4/3)π(18/π).
This further simplifies to V = 24 cm^3.
Therefore, the volume of the sphere is 24 cm^3.