Supposed to fear cylinder and cone all share the same radius in 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? Why is the correct answer 3 cones = 1 cylinder and NOT 1 cylinder = 2 cones + 1 sphere
The correct answer is 3 cones = 1 cylinder, not 1 cylinder = 2 cones + 1 sphere for the given scenario.
To understand why this is the case, let's compare the volume formulas for the sphere, cylinder, and cone:
Volume of a sphere: V_s = (4/3)πr^3
Volume of a cylinder: V_c = πr^2h
Volume of a cone: V_co = (1/3)πr^2h
In this scenario, both the cylinder and the cone share the same radius and height:
Radius (r) = Radius of the cone = Radius of the cylinder
Height (h) = 2 * radius
To find the relationship between the volume formulas of the sphere, cylinder, and cone, we need to compare their volumes using the given information.
Let's start with the relationship between the volumes of a cone and a cylinder:
V_co = (1/3)πr^2h
V_c = πr^2h
Dividing V_co by V_c, we get:
V_co / V_c = ((1/3)πr^2h) / (πr^2h)
V_co / V_c = (1/3)
This means that the volume of a cone is one-third of the volume of a cylinder when they share the same radius and height.
Now, let's compare the relationship between the volumes of three cones and one cylinder:
3 * V_co = 3 * ((1/3)πr^2h) (since the volume of one cone is (1/3)πr^2h)
3 * V_co = πr^2h
Comparing this to the volume of a cylinder:
V_c = πr^2h
We can see that 3 cones have the same volume as one cylinder.
Therefore, the correct answer is 3 cones = 1 cylinder.