Question

A cylinder with the altitude h is inscribed in a cone with the altitude H and the radius of the base R. Find the radius of the base of the cylinder.

Answer 1

Let one angle is $\theta$

In $\Delta ABC$,

$\tan \theta =\frac{R}{H}⟶\left(i\right)$

In $\Delta ADE$,

$\cot \theta =\frac{H-R}{r}$

$\Rightarrow \tan \theta =\frac{r}{H-R}⟶\left(ii\right)$

From $\left(i\right)$ & $\left(ii\right)$

$\frac{R}{H}=\frac{r}{H-h}\Rightarrow r=R\left(\frac{H-h}{H}\right)$