Suppose a sphere, cylinder, and cone all share the same radius. The cylinder and cone also share the same height, which is twice the radius. Which of the following is true about the relationship between the volume formulas for the sphere, cylinder, and cone?(1 point)
Responses
cone = cylinder – sphere
cone = cylinder – sphere
cylinder = sphere – cone
cylinder = sphere – cone
sphere = cylinder + cone
sphere = cylinder + cone
cone = sphere – cylinder
cone = sphere - cylinder
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
Let's use the relationship between the volumes of a cylinder and sphere to solve the problem.
The volume of a cylinder is given by the formula V_cylinder = π * r^2 * h, where r is the radius and h is the height.
The height of the cylinder is equal to the sphere's diameter, which means it is twice the radius. So, h = 2r.
Given that the volume of the cylinder is 36 cm^3, we can substitute the values into the formula:
36 = π * r^2 * (2r)
Now, let's solve for the radius of the cylinder:
36 = 2π * r^3
r^3 = 36 / (2π)
r^3 = 18 / π
r ≈ 1.92 cm
Now, let's find the volume of the sphere using the formula V_sphere = (4/3) * π * r^3:
V_sphere ≈ (4/3) * π * (1.92)^3
≈ (4/3) * π * 7.21
≈ 30.27 cm^3
So, the volume of the sphere is approximately 30.27 cm^3.
The correct answer choice is:
30.27 cm3
30.27 cm cubed