Volume of Cones, Cylinders, and Spheres Quick Check
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 cm?. 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?
First, we need to find the height of the cylinder. The formula for the volume of a cylinder is V = πr²h, where V is the volume, r is the radius, and h is the height. Given that the volume of the cylinder is 36 cm?, let's substitute the given values into the formula:
36 = πr²h
Since the radius of the sphere is the same as the cylinder's, we can substitute r for the radius of the sphere.
36 = π(r)²h
We are also given that the height of the cylinder is equal to the sphere's diameter, which means the height is twice the radius of the sphere. Let's substitute 2r for h.
36 = π(r)²(2r)
Now, let's simplify the expression:
36 = π(r²)(2r)
36 = 2πr³
To find the volume of the sphere, we can use the formula V = (4/3)πr³, where V is the volume, and r is the radius. Comparing this formula to the simplified equation above, we can see that the volume of the sphere is (4/3) times the volume of the cylinder. Therefore, the volume of the sphere is:
V_sphere = (4/3)(36)
V_sphere = 48 cm?