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)
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 and the height of the cylinder is equal to the sphere's diameter, we can say that h = 2r.
Substituting h = 2r into the formula for the volume of a cylinder, we have:
36 = πr^2(2r)
Simplifying the equation:
36 = 2πr^3
Dividing both sides by 2π:
18/π = r^3
Taking the cube root of both sides:
r = (18/π)^(1/3)
Substituting r back into the formula for the volume of a sphere:
V = (4/3)π(18/π)^(1/3)^3
V = (4/3)π(18/π)
V = (4/3)(18)
V = 24 cm^3
Therefore, the volume of the sphere is 24 cm^3.