Question

If the volumes of two cones are in the ratio of 1 : 4 and their diameters are in the ratio of 4 : 5, find the ratio of their heights.

Answer 1

Let r and R be the base radii, h and H be the heights , v and 

We have,

`(2r)/(2R) = 4/5 or r/R = 4/5`       ............(i)

and 

`v/V = 1/4`

`rArr ((1/3pir^2h))/((1/3piR^2H))=1/4`

`rArr ((r^2h))/((R^2H)) = 1/4`

`rArr (r/R)^2xx h/H = 1/4`

`rArr (4/5)^2xxh/H = 1/4`     [Using (i)]

`rArr (16/25)xx h/H=1/4`

`rArr h/H = (1xx25)/(4xx16)`

`rArr h/H = 25/64`

`therefore h: H = 25:64`

So, the ratio of their heights is 25:64.