The radius (in cm) of the right circular cyli | vedclass

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(B) Let the radius of the cone be $R = 10 \ cm$ and its height be $H$. Let the radius of the inscribed cylinder be $r$ and its height be $h$.By simila

Vedclass · Accepted Answer

$5$

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(B) Let the radius of the cone be $R = 10 \ cm$ and its height be $H$. Let the radius of the inscribed cylinder be $r$ and its height be $h$.By similar triangles,$\frac{H-h}{r} = \frac{H}{R}$,which implies $h = H(1 - \frac{r}{R})$.The curved surface area $S$ of the cylinder is $S = 2\pi rh = 2\pi r H(1 - \frac{r}{R}) = 2\pi H(r - \frac{r^2}{R})$.To maximize $S$,we differentiate with respect to $r$ and set to zero: $\frac{dS}{dr} = 2\pi H(1 - \frac{2r}{R}) = 0$.This gives $1 - \frac{2r}{R} = 0$,so $r = \frac{R}{2}$.Given $R = 10 \ cm$,we have $r = \frac{10}{2} = 5 \ cm$.

The radius (in $cm$) of the right circular cylinder of greatest curved surface area which can be inscribed in a given cone with radius $10 \ cm$ is

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