Suppose a sphere 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 that 3 cones have the same volume as 1 cylinder. This is because the volume of a cone is given by the formula (1/3)πr^2h, where r is the radius and h is the height. Since the height of the cone is equal to twice the radius, we can rewrite the formula as (1/3)πr^2(2r) = (2/3)πr^3.
On the other hand, the volume of a cylinder is given by the formula πr^2h, where r is the radius and h is the height. In this case, the height of the cylinder is also equal to twice the radius, so the formula becomes πr^2(2r) = 2πr^3.
Therefore, the volume of the cone (2/3)πr^3 is exactly two-thirds of the volume of the cylinder 2πr^3. So, the correct answer is that 3 cones have the same volume as 1 cylinder.
It is not true that 1 cylinder equals 2 cones + 1 sphere. This is because the volume formula for a sphere is different from the volume formulas for a cylinder and a cone. The volume of a sphere is given by the formula (4/3)πr^3. Therefore, the relationship between the volumes of a sphere, cylinder, and cone cannot be expressed in the form 1 cylinder = 2 cones + 1 sphere.