Volume of a cylinder is in joint variation with squqre of the length of radius of base and its height . Ratio of radii of bases of two of cylinders `2:3` and ratio of their height is `5:4` , then find the ratio of their volumes .

Explanation

Let the volume of the cylinder = V,
Radius of the base =R and height =H .
As per question, V `prop R^2 H rArr V k.R^2H ("where "k ne0=` variation cosntant ).
Now , the volume of the cylinders be `V_1 and V_2 ` and their radii are 2r and 3r respectively and heights are 5h and 4h respectively [`because ` ratio of radii =2:3 and ratio height =5:4 ]
`therefore` from (1) we get , `V_1 =k.(2r)^2 .5h =20kr^2 h [because R =2 and H =5h] `
and `V_2 =k.(3r)^2 .4h =36 kr^2 g [because R =2r and H =4h ] `
`therefore (V_1)/(V_2) =(20kr^2h)/(36kr^2h ) =5/9 therefore v_1 :V_2 =5:9 `
`therefore` The ratio of the volumes of the cylinders is `5:9` .