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Question-148171




Question Number 148171 by ajfour last updated on 25/Jul/21
Commented by ajfour last updated on 25/Jul/21
This bird caged inside this   tetrahedron has to start from  origin, touch each centroidal  points of the faces and get  back. If tetrahedron is regular  with edge a, find minimum   distance of the journey.  (worm holes not allowed  :)
$${This}\:{bird}\:{caged}\:{inside}\:{this}\: \\ $$$${tetrahedron}\:{has}\:{to}\:{start}\:{from} \\ $$$${origin},\:{touch}\:{each}\:{centroidal} \\ $$$${points}\:{of}\:{the}\:{faces}\:{and}\:{get} \\ $$$${back}.\:{If}\:{tetrahedron}\:{is}\:{regular} \\ $$$${with}\:{edge}\:{a},\:{find}\:{minimum}\: \\ $$$${distance}\:{of}\:{the}\:{journey}. \\ $$$$\left({worm}\:{holes}\:{not}\:{allowed}\:\::\right) \\ $$
Answered by mr W last updated on 25/Jul/21
Commented by mr W last updated on 25/Jul/21
BC=CD=DE=(a/3)  AB=EA=(2/3)×(((√3)a)/2)=(((√3)a)/3)  mininum path=2×(((√3)a)/3)+3×(a/3)=(((3+2(√3))a)/3)  ≈2.155a
$${BC}={CD}={DE}=\frac{{a}}{\mathrm{3}} \\ $$$${AB}={EA}=\frac{\mathrm{2}}{\mathrm{3}}×\frac{\sqrt{\mathrm{3}}{a}}{\mathrm{2}}=\frac{\sqrt{\mathrm{3}}{a}}{\mathrm{3}} \\ $$$${mininum}\:{path}=\mathrm{2}×\frac{\sqrt{\mathrm{3}}{a}}{\mathrm{3}}+\mathrm{3}×\frac{{a}}{\mathrm{3}}=\frac{\left(\mathrm{3}+\mathrm{2}\sqrt{\mathrm{3}}\right){a}}{\mathrm{3}} \\ $$$$\approx\mathrm{2}.\mathrm{155}{a} \\ $$
Commented by ajfour last updated on 25/Jul/21
I had not thought it wd b that  easy, thank you sir, correct  n great image.
$${I}\:{had}\:{not}\:{thought}\:{it}\:{wd}\:{b}\:{that} \\ $$$${easy},\:{thank}\:{you}\:{sir},\:{correct} \\ $$$${n}\:{great}\:{image}. \\ $$

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