Suppose a sphere, cylinder, and cone all share the same radius and 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
sphere - cone = cylinder
3 cones = 1 cylinder
sphere = cylinder + cone
1 cylinder = 2 cones + 1 sphere
The correct answer is sphere = cylinder + cone.
The volume of a sphere is 48 m^3 . What is the volume of a cylinder if its radius is the same as the sphere’s and its height is equal to the sphere’s diameter?(1 point) ___m^3
The diameter of the sphere is twice its radius. Therefore, if the diameter of the sphere is D, the height of the cylinder would also be D.
The volume of a sphere is given by the formula V_sphere = (4/3)πr^3, where r is the radius. In this case, V_sphere = 48 m^3.
To find the radius, we can rearrange the formula:
48 = (4/3)πr^3
144/π = r^3
r = (144/π)^(1/3)
Now that we have the radius, we can calculate the volume of the cylinder using the formula V_cylinder = πr^2h, where r is the radius and h is the height. In this case, h = D.
V_cylinder = π[(144/π)^(1/3)]^2 × D
V_cylinder = π[144/π]^(2/3) × D^3/DD
Since D^3/DD = D^2, the formula becomes:
V_cylinder = π[144/π]^(2/3) × D^2
Therefore, the volume of the cylinder is π[144/π]^(2/3) × D^2 m^3.
So what is the answer
Apologies for the confusion. The volume of the cylinder with a radius equal to the sphere's radius and a height equal to the sphere's diameter is π × r^2 × h, where r is the radius and h is the height.
Since the diameter of the sphere is twice the radius, let's assume the radius of the sphere is r. Therefore, the diameter of the sphere is 2r, and the height of the cylinder is 2r.
Substituting these values into the formula, we get:
V_cylinder = π × r^2 × 2r
V_cylinder = 2πr^3
So, the volume of the cylinder is 2πr^3 cubic units.
I still don't understand and why did you change the answer?
Apologies for the confusion earlier. I made an error in my previous response.
Given that the volume of the sphere is 48 m^3 and the height of the cylinder is equal to the sphere's diameter, which is also twice the radius, we can proceed as follows:
We know that the volume of a sphere is given by the formula V_sphere = (4/3)πr^3, where r is the radius. In this case, V_sphere is given as 48 m^3.
We can rearrange the formula to solve for the radius:
48 = (4/3)πr^3
36/π = r^3
r = (36/π)^(1/3)
Now, to calculate the volume of the cylinder, we can use the formula V_cylinder = πr^2h, where r is the radius and h is the height. In this case, the height of the cylinder is equal to the diameter of the sphere, which is twice the radius.
h = 2r
Substituting the value of r, we have:
h = 2 * (36/π)^(1/3)
Now we can calculate the volume of the cylinder:
V_cylinder = πr^2h
V_cylinder = π * [(36/π)^(1/3)]^2 * [2 * (36/π)^(1/3)]
Simplifying this expression will give us the volume of the cylinder.
I apologize for the mistake in my previous response and any confusion it may have caused.