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
54 cm3
54 cm cubed
12 cm3
12 cm cubed
18 cm3
18 cm cubed
The volume of a cylinder is given by the formula Vcylinder = πr^2h, where r is the radius and h is the height.
Given that the volume of the cylinder is 36 cm^3, we can solve for the radius by using the formula for volume:
36 = πr^2h
Since the height of the cylinder is equal to the sphere's diameter, we can substitute h = 2r:
36 = πr^2(2r)
36 = 2πr^3
Now, let's compare this equation to the formula for the volume of a sphere:
Vsphere = (4/3)πr^3
We can see that the volume of the cylinder is half of the volume of the sphere, because the coefficient of r^3 in the equation for the cylinder is half of the coefficient of r^3 in the equation for the sphere. Therefore, the volume of the sphere is twice the volume of the cylinder.
Since the volume of the cylinder is 36 cm^3, the volume of the sphere is:
Vsphere = 2 * 36 = 72 cm^3
So, the correct answer is 72 cm^3.