There are two cones. The curved surface area of one is twice that of the other. The slant height of the latter is twice that of the former. Find the ratio of their radii.

Let curved surface area of 1st cone be twice than that of 2nd cone.

For 1st cone,

⇒ Slant height (l₁) = l

⇒ Radius = r₁

⇒ Curved surface area (C₁) = 2C ………(1)

For 2nd cone,

⇒ Slant height (l₂) = 2l

⇒ Radius = r₂

⇒ Curved surface area (C₂) = C ………(2)

Dividing equation (1) by (2), we get :

$\Rightarrow \dfrac{\text{CSA of 1st cone}}{\text{CSA of 2nd cone}} = \dfrac{2C}{C} \\[1em] \Rightarrow \dfrac{πr$1l1}{πr2l2} = \dfrac{2C}{C} \\[1em] \Rightarrow \dfrac{πr1.l}{πr2.2l} = 2 \\[1em] \Rightarrow \dfrac{r1}{2r2} = 2 \\[1em] \Rightarrow \dfrac{r1}{r2} = \dfrac{4}{1}.

⇒ r₁ : r₂ = 4 : 1.

Hence, ratio of radii = 4 : 1.