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Question Number 118104 by Lordose last updated on 15/Oct/20
1.)  A right circular cone is circumscribed  about a sphere of radius(r).  If d is the   distance from the center of the sphere  to the vertex of the cone, show that the  volume of the cone,V=((𝛑r^2 (r+d)^2 )/(3(d−r))).  2.)  Find the vertical angle of the cone when  it′s volume is minimum.
$$\left.\mathrm{1}.\right) \\ $$$$\mathrm{A}\:\mathrm{right}\:\mathrm{circular}\:\mathrm{cone}\:\mathrm{is}\:\mathrm{circumscribed} \\ $$$$\mathrm{about}\:\mathrm{a}\:\mathrm{sphere}\:\mathrm{of}\:\mathrm{radius}\left(\boldsymbol{\mathrm{r}}\right).\:\:\mathrm{If}\:\boldsymbol{\mathrm{d}}\:\mathrm{is}\:\mathrm{the}\: \\ $$$$\mathrm{distance}\:\mathrm{from}\:\mathrm{the}\:\mathrm{center}\:\mathrm{of}\:\mathrm{the}\:\mathrm{sphere} \\ $$$$\mathrm{to}\:\mathrm{the}\:\mathrm{vertex}\:\mathrm{of}\:\mathrm{the}\:\mathrm{cone},\:\mathrm{show}\:\mathrm{that}\:\mathrm{the} \\ $$$$\mathrm{volume}\:\mathrm{of}\:\mathrm{the}\:\mathrm{cone},\boldsymbol{\mathrm{V}}=\frac{\boldsymbol{\pi\mathrm{r}}^{\mathrm{2}} \left(\boldsymbol{\mathrm{r}}+\boldsymbol{\mathrm{d}}\right)^{\mathrm{2}} }{\mathrm{3}\left(\boldsymbol{\mathrm{d}}−\boldsymbol{\mathrm{r}}\right)}. \\ $$$$\left.\mathrm{2}.\right) \\ $$$$\boldsymbol{\mathrm{F}}\mathrm{ind}\:\mathrm{the}\:\mathrm{vertical}\:\mathrm{angle}\:\mathrm{of}\:\mathrm{the}\:\mathrm{cone}\:\mathrm{when} \\ $$$$\mathrm{it}'\mathrm{s}\:\mathrm{volume}\:\mathrm{is}\:\mathrm{minimum}. \\ $$
Commented by 1549442205PVT last updated on 15/Oct/20
This is repeated question Q118069
$$\mathrm{This}\:\mathrm{is}\:\mathrm{repeated}\:\mathrm{question}\:\mathrm{Q118069} \\ $$

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