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 24 cm3 24 cm cubed 54 cm3 54 cm cubed 12 cm3 12 cm cubed
To solve this problem, we can use the relationship between the volumes of a cylinder and a sphere.
The volume of a cylinder is given by the formula: Vcylinder = πr²h, where r is the radius and h is the height.
Given that the volume of the cylinder is 36 cm³, we have: 36 = πr²h
We are also given that the height of the cylinder is equal to the diameter of the sphere. The diameter of a sphere is twice its radius, so the height of the cylinder is 2r.
Substituting this value into the equation: 36 = πr²(2r)
Simplifying the equation: 36 = 2πr³
Dividing both sides by 2π: r³ = 18
Taking the cube root of both sides: r ≈ 2.620
Now, let's find the volume of the sphere using the formula: Vsphere = (4/3)πr³
Substituting the value of r: Vsphere ≈ (4/3)π(2.620)³ ≈ 24.04 cm³
Therefore, the volume of the sphere is approximately 24 cm³. Answer: 24 cm³