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A cone and a hemisphere have equal bases and equal volumes. What is the ratio of their heights?
- a)1 : 2
- b)2 : 1
- c)√2 : 1
- d)4 : 1
Correct answer is option 'B'. Can you explain this answer?
Most Upvoted Answer
A cone and a hemisphere have equal bases and equal volumes. What is th...
Cone and hemisphere have equal base means equal radii, of R cm,
Height of the cone be H cm.
Height of hemisphere = R cm.
Volume of cone = volume of hemisphere
H/R = 2/1
Height of the cone be H cm.
Height of hemisphere = R cm.
Volume of cone = volume of hemisphere
H/R = 2/1
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A cone and a hemisphere have equal bases and equal volumes. What is th...
The ratio of heights of a cone and a hemisphere can be determined by comparing their volumes and bases.
- **Volume of a Cone and a Hemisphere**: The volume of a cone is given by Vcone = (1/3)πr^2h, where r is the radius of the base and h is the height. The volume of a hemisphere is Vhemisphere = (2/3)πr^3, where r is the radius of the base.
- **Equal Volumes**: Since the volumes of the cone and hemisphere are equal, we can set their volume equations equal to each other:
(1/3)πr^2h = (2/3)πr^3
Simplifying, we get h = 2r.
- **Equal Bases**: Given that the cone and hemisphere have equal bases, the radius of the cone base is equal to the radius of the hemisphere base, i.e., rcone = rhemisphere.
- **Ratio of Heights**: From the equation h = 2r, we can see that the height of the cone is twice the radius of its base. Since the radius of the cone base is equal to the radius of the hemisphere base, the height of the cone is twice the height of the hemisphere. Therefore, the ratio of their heights is 2 : 1.
Therefore, the correct answer is **option B) 2 : 1**, indicating that the height of the cone is twice the height of the hemisphere when they have equal bases and volumes.
- **Volume of a Cone and a Hemisphere**: The volume of a cone is given by Vcone = (1/3)πr^2h, where r is the radius of the base and h is the height. The volume of a hemisphere is Vhemisphere = (2/3)πr^3, where r is the radius of the base.
- **Equal Volumes**: Since the volumes of the cone and hemisphere are equal, we can set their volume equations equal to each other:
(1/3)πr^2h = (2/3)πr^3
Simplifying, we get h = 2r.
- **Equal Bases**: Given that the cone and hemisphere have equal bases, the radius of the cone base is equal to the radius of the hemisphere base, i.e., rcone = rhemisphere.
- **Ratio of Heights**: From the equation h = 2r, we can see that the height of the cone is twice the radius of its base. Since the radius of the cone base is equal to the radius of the hemisphere base, the height of the cone is twice the height of the hemisphere. Therefore, the ratio of their heights is 2 : 1.
Therefore, the correct answer is **option B) 2 : 1**, indicating that the height of the cone is twice the height of the hemisphere when they have equal bases and volumes.
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