Question and Answers Forum

All Questions      Topic List

Geometry Questions

Previous in All Question      Next in All Question      

Previous in Geometry      Next in Geometry      

Question Number 53664 by ajfour last updated on 24/Jan/19

Commented by ajfour last updated on 24/Jan/19

The bases of the cones touch each  other pairwise. Find the radius of  the sphere tangent to all the three  cones and to the plane passing  through their vertices. (semi-vertical  angles of the cones is α/2)

$${The}\:{bases}\:{of}\:{the}\:{cones}\:{touch}\:{each} \\ $$$${other}\:{pairwise}.\:{Find}\:{the}\:{radius}\:{of} \\ $$$${the}\:{sphere}\:{tangent}\:{to}\:{all}\:{the}\:{three} \\ $$$${cones}\:{and}\:{to}\:{the}\:{plane}\:{passing} \\ $$$${through}\:{their}\:{vertices}.\:\left({semi}-{vertical}\right. \\ $$$$\left.{angles}\:{of}\:{the}\:{cones}\:{is}\:\alpha/\mathrm{2}\right) \\ $$

Answered by mr W last updated on 24/Jan/19

Commented by mr W last updated on 24/Jan/19

a=(((√3)×2r)/2)×(2/3)=((2(√3)r)/3)  (a/(tan α))=(R/(sin α))+R  ⇒R=((cos α)/(1+sin α))×a  ⇒R=((2(√3) cos α r)/(3(1+sin α)))

$${a}=\frac{\sqrt{\mathrm{3}}×\mathrm{2}{r}}{\mathrm{2}}×\frac{\mathrm{2}}{\mathrm{3}}=\frac{\mathrm{2}\sqrt{\mathrm{3}}{r}}{\mathrm{3}} \\ $$$$\frac{{a}}{\mathrm{tan}\:\alpha}=\frac{{R}}{\mathrm{sin}\:\alpha}+{R} \\ $$$$\Rightarrow{R}=\frac{\mathrm{cos}\:\alpha}{\mathrm{1}+\mathrm{sin}\:\alpha}×{a} \\ $$$$\Rightarrow{R}=\frac{\mathrm{2}\sqrt{\mathrm{3}}\:\mathrm{cos}\:\alpha\:{r}}{\mathrm{3}\left(\mathrm{1}+\mathrm{sin}\:\alpha\right)} \\ $$

Commented by ajfour last updated on 25/Jan/19

or from half of uppermost   quadrilateral,  tan ((π/4)−(α/2))=(R/a)  with acos 30°=r  R = ((2r)/(√3))tan ((π/4)−(α/2)).

$${or}\:{from}\:{half}\:{of}\:{uppermost}\: \\ $$$${quadrilateral},\:\:\mathrm{tan}\:\left(\frac{\pi}{\mathrm{4}}−\frac{\alpha}{\mathrm{2}}\right)=\frac{{R}}{{a}} \\ $$$${with}\:{a}\mathrm{cos}\:\mathrm{30}°={r} \\ $$$${R}\:=\:\frac{\mathrm{2}{r}}{\sqrt{\mathrm{3}}}\mathrm{tan}\:\left(\frac{\pi}{\mathrm{4}}−\frac{\alpha}{\mathrm{2}}\right). \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com