The radii of the ends of a frustum of a cone are r₁ and r₂. If l is the slant height of the frustum, then the slant height of the cone of which the frustum is apart, is ________.

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Solution

Let the slant height of the cone be L units.

Here, ∆OAB $~$ ∆OCD

$∴ \frac{OB}{OD} = \frac{AB}{CD} \Rightarrow \frac{L - l}{L} = \frac{r_{1}}{r_{2}} \Rightarrow 1 - \frac{l}{L} = \frac{r_{1}}{r_{2}}$
$\Rightarrow \frac{l}{L} = 1 - \frac{r_{1}}{r_{2}} \Rightarrow \frac{l}{L} = \frac{r_{2} - r_{1}}{r_{2}} \Rightarrow L = (\frac{r_{2}}{r_{2} - r_{1}}) l$
The radii of the ends of a frustum of a cone are r₁ and r₂. If l is the slant height of the frustum, then the slant height of the cone of which the frustum is apart, is $(\frac{r_{2}}{r_{2} - r_{1}}) l$ .

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