If the lateral areas of two similar cylinders are 50pi cm sq and 18pi cm sq and the volume of the smaller cylinder is 54pi cm cubed what is the volume of the larger cylinder?

The two cylinders are similar - they are of corresponding shapes but one is x times bigger than the other. If a cylinder is x times bigger in a single dimension, it will be x2 times greater in measurements involving two dimensions (i.e. area) and x3 times bigger in three dimensions (volume). Here, the ratio in two dimensions is 50pi / 18pi. This cancels to 25/9. So in terms of area, one is 25/9 times larger. x2 = 25/9

So x = sqrt25/sqrt9 = 5/3

and x3 = 53/33 = 125/27 (If you prefer, one can just write (25/9)3/2 = 125/27) In terms of volume, then, the big cylinder is 125/27 times bigger. Multiply this ratio by the volume of the small cylinder... 54pi x 125/27 cancels to 2pi x 125 = 250pi cm3. ...and you have the volume of the big cylinder. (It's tidiest to leave the answer in terms of pi.) This is just a question about scale factors, and the identity of the objects as cylinders is irrelevant, because they are similar. The way in which the figures neatly cancel without producing any horrendous decimals suggests that this question came from a textbook or an exam paper.