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Question
The surface areas of two spheres are in the ratio of 4 : 25. Find the ratio of their volumes.
Solution
Let the radii of the two spheres be r and R
As,
`"Surface area of the first sphere"/"Surface area of the second sphere" = 4/25`
`=> (4pi"r"^2)/(4pi"R"^2) = 4/25`
`=> ("r"/R)^2 = 4/25`
`=> r/R = sqrt(4/25)`
`=> r/R = 2/5` ..........(i)
Now,
The ratio of the Volumes of the two spheres `= ("Volume of the first sphere" )/("Volume of the second sphere")`
`= ((4/3pi"r"^3))/((4/3pi"R"^3))`
`= ("r"/"R")^3`
`=(2/5)^3` [Using (i)]
`=8/125`
= 8 : 125
So, the ratio of the volumes of the given spheres is 8 : 125.
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