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

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