Question

The surface areas of two spheres are in the ratio of 4 : 25. Find the ratio of their volumes.

Answer 1

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.