Use the relationship among the formulas for the volume 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 it’s radius is the same as the cylinders and the height of the cylinder is equal to the spheres diameter.
A 18 cm³
B 24 cm³
C 54 cm³
D 12 cm³
First, let's find the radius of the cylinder using the formula for the volume of a cylinder: V_cylinder = πr^2h, where V_cylinder is the volume of the cylinder, r is the radius, and h is the height.
Since the volume of the cylinder is 36 cm³ and h is equal to the sphere's diameter, let's assume h = 2r.
36 = πr^2(2r)
Divide both sides by 2π:
18/π = r^3
Take the cube root of both sides:
r ≈ (18/π)^(1/3) ≈ 1.835 cm
Now that we have the radius of the cylinder, which is the same as the radius of the sphere, we can calculate the volume of the sphere using the formula: V_sphere = (4/3)πr^3.
V_sphere = (4/3)π(1.835)^3
V_sphere ≈ 24.05 cm³
Therefore, the volume of the sphere is approximately 24 cm³, so the answer is B) 24 cm³.