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
18 cm3
18 cm cubed
54 cm3
54 cm cubed
24 cm3
24 cm cubed
12 cm3
12 cm cubed
To solve this problem, we need to use the relationship among the formulas for the volumes of cones, cylinders, and spheres.
The formula for the volume of a cylinder is Vcylinder = πr^2h, where r is the radius and h is the height of the cylinder.
The formula for the volume of a sphere is Vsphere = (4/3)πr^3, where r is the radius of the sphere.
Given that the volume of the cylinder is 36 cm3 and the height of the cylinder is equal to the sphere's diameter, we can calculate the radius of the cylinder.
Using the formula for the volume of the cylinder, we have 36 = πr^2h.
Since the height of the cylinder is equal to the sphere's diameter, we have h = 2r.
Substituting this value into the equation, we have 36 = πr^2(2r).
Simplifying the equation, we get 36 = 2πr^3.
Dividing both sides of the equation by 2π, we have r^3 = 18.
Taking the cube root of both sides, we find that r = ∛18.
Now that we know the radius of the sphere, we can calculate the volume of the sphere using the formula Vsphere = (4/3)πr^3.
Substituting the value of r, we have Vsphere = (4/3)π(∛18)^3.
Simplifying this expression, we get Vsphere = (4/3)π(18∛2).
So, the volume of the sphere is 24 cm3 (rounded to the nearest whole number).