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
24 cm3
24 cm cubed
12 cm3
12 cm cubed
54 cm3
54 cm cubed
18 cm3
The formula for the volume of a cylinder is V = πr^2h, where r is the radius and h is the height.
The radius of the cylinder is the same as the radius of the sphere.
The height of the cylinder is equal to the sphere's diameter, which means it is twice the radius of the sphere.
So, the height of the cylinder is 2r.
The volume of the cylinder is given as 36 cm^3.
Plugging in these values into the formula, we have 36 = πr^2 * 2r.
Simplifying, we get 36 = 2πr^3.
Dividing both sides of the equation by 2π, we get r^3 = 18.
Taking the cube root of both sides, we get r = 3.
So, the radius of the sphere is 3 cm.
Now, using the formula for the volume of a sphere, V = (4/3)πr^3, we can calculate the volume of the sphere.
Substituting r = 3 into the formula, we have V = (4/3)π(3)^3 = 36π cm^3.
Approximating π to 3.14, we have V ≈ 36(3.14) = 113.04 cm^3.
Therefore, the volume of the sphere is approximately 113.04 cm^3.