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Question
A cone and a hemisphere have equal bases and equal volumes the ratio of their heights is
Options
1 : 2
2 : 1
4 : 1
- \[\sqrt{2}\] : 1
Solution
In the given problem, we are given a cone and a hemisphere which have equal bases and have equal volumes. We need to find the ratio of their heights.
So,
Let the radius of the cone and hemisphere be x cm.
Also, height of the hemisphere is equal to the radius of the hemisphere.
Now, let the height of the cone = h cm
So, the ratio of the height of cone to the height of the hemisphere = `h/x`
Here, Volume of the hemisphere = volume of the cone
`(2/3) pi r_h^3 = (1/3) pi r_c ^2 h `
`(2/3) pi (x)^3 = (1/3) pi (x)^2 h`
`(2/3) (x) = (1/3) h`
`2x = 1h`
`h/x = 2/1`
Therefore, the ratio of the heights of the cone and the hemisphere is 2 : 1.
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