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A right circular cone, a right circular cylinder and a hemisphere have the same radius. Heights of the cone and the cylinder are equal to their diameters. The ratio of the volumes of cone, cylinder and hemisphere
  • a)
    1 : 3 : 1
  • b)
    2 : 1 : 3
  • c)
    3 : 2 : 1
  • d)
    1 : 2 : 3
Correct answer is option 'A'. Can you explain this answer?
Most Upvoted Answer
A right circular cone, a right circular cylinder and a hemisphere hav...
Given:
- The radius of the cone, cylinder, and hemisphere is the same.
- The height of the cone and cylinder is equal to their diameters.

To find:
The ratio of the volumes of the cone, cylinder, and hemisphere.

Solution:
Let's assume the radius and height of the cone, cylinder, and hemisphere to be 'r' and 'h' respectively.

Volume of the Cone:
The volume of a cone is given by the formula Vcone = (1/3)πr²h.
Given that the radius is 'r' and the height is 'h', the volume of the cone can be written as Vcone = (1/3)πr²(2r) = (2/3)πr³.

Volume of the Cylinder:
The volume of a cylinder is given by the formula Vcylinder = πr²h.
Given that the radius is 'r' and the height is 'h', the volume of the cylinder can be written as Vcylinder = πr²(2r) = 2πr³.

Volume of the Hemisphere:
The volume of a hemisphere is given by the formula Vhemisphere = (2/3)πr³.
Given that the radius is 'r', the volume of the hemisphere can be written as Vhemisphere = (2/3)πr³.

Ratio of the Volumes:
To find the ratio of the volumes, we can compare the volumes of the cone, cylinder, and hemisphere.
Vcone : Vcylinder : Vhemisphere = (2/3)πr³ : 2πr³ : (2/3)πr³ = (2/3)πr³ : (6/3)πr³ : (2/3)πr³ = 1 : 3 : 1

Answer:
Therefore, the ratio of the volumes of the cone, cylinder, and hemisphere is 1 : 3 : 1, which is option 'A'.
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Community Answer
A right circular cone, a right circular cylinder and a hemisphere hav...
Radius of cone = Radius of cylinder = Radius of hemisphere = r (because it is given that the cone, cylinder and hemisphere have the same radius)
Height of cone = Height of cylinder = 2r (because it is given that heights of the cylinder and the cone are the same as their diameters)
Ratio of their volumes
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