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

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 Link

Everest2017 Everest2017 · Answer 2 · 2014-08-09T16:58:14-04:00

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