What is the ratio of the volume of a cylinder and a hemisphere of equal radii and equal height?
Answers
ANSWER:__✍️
The volume of a cone, V(cone), is given by the following formula:
V(cone) = (1/3)Bh, where B is the area of the circular base and h is the height of the cone. The area B of the base is given by the formula B = πr², where r is the radius of the circular base; Therefore, substituting for B, the volume of a cone becomes:
V(cone) = (1/3)πr²h
The volume of a cylinder, V(cylinder), is given by the following formula:
V(cylinder) = Bh, where B is the area of each of the two circular bases and h is the height of the cylinder. The area B of each base is again given by the formula B = πr², where r is the radius of each circular base; Therefore, substituting for B, the volume of a cylinder becomes:
V(cylinder) = πr²h
Since a given cone and a cylinder have the same radius r and height h, then the ratio of their volumes is:
V(cone)/V(cylinder) = [(1/3)πr²h]/(πr²h)
= (1/3)(π/π)(r²/r²)(h/h)
= (1/3)(1)(1)(1)
= 1/3
Therefore, for a cone which has the same radius and height as a cylinder, we see that the volume of the cone is one-third (1/3) the volume of the cylinder.