The diameters of two cones are equal. If their slant heights are in the ratio 5 : 4, find the ratio of their curved surface areas.

Given, ratio of slant height = 5 : 4

Let slant height of 1st cone be 5x cm and 2nd cone be 4x cm.

For 1st cone,

⇒ Diameter = d₁

⇒ Radius = r₁

⇒ Slant height (l₁) = 5x

For 2nd cone,

⇒ Diameter = d₂

⇒ Radius = r₂

⇒ Slant height (l₂) = 4x

Given,

⇒ d₁ = d₂

∴ r₁ = r₂.

$\Rightarrow \dfrac{\text{CSA of 1st cone}}{\text{CSA of 2nd cone}} = \dfrac{πr$1l1}{πr2l2} \\[1em] = \dfrac{l1}{l2} = \dfrac{5x}{4x} \\[1em] = \dfrac{5}{4} = 5 : 4.

Hence, ratio of curved surface areas = 5 : 4.