Question

two similar cones have surface areas in the ration 4:9. find the ration of their lengths and their volumes

Answer 1

Well, the ratio of area is (Ratio of length)^2. So we work backwards. 

ROL = [sqrt] 4:9 
= 2:3 

and its the same for height, because height is a length. 

NB: Ratio of volume = (Ratio of length)^3

Answer 2

Cone : r radius of bottom circle,  L lateral length,  h = perpendicular height

   Total surface area = π r² ( 1 + L / r )             Volume = 1/3 π r² h
   L²  = r² + h²

   Similar cones =>  r1 : r2  = L1: L2  =  h1 : h2
                         =>  L1 / r1 = L2 / r2     => 1+ L1/r1  = 1 + L2/r2
   S1 : S2 = 2² : 3²  = r1² : r2²
   r1 : r2 = 2 : 3  also  h1 : h2 = 2 : 3
 
   Volumes V1 : V2 =  r1² h1 / r2² h2  = (2/3)² 2/3  = 8 / 9