I'm trying to solve this problem:

Two wheels are one next to another and linked by a conveyor belt. How to express the length of the radius of the smaller wheel in terms of the length of the belt? Note that the size of the radius of A is three times the radius of B.

Sketch of the problem (not in scale)


Let the length of the belt be $b$.

enter image description here

\begin{align} d&=\sqrt{(4r)^2-(2r)^2} =2\sqrt{3}r ,\\ \phi&=\arccos\tfrac{2r}{4r}=\tfrac\pi3 ,\\ b&=2\,((\pi-\phi)\cdot3r+d+\phi\,r) \\ &=2(2\pi r+2\sqrt{3}r+\tfrac\pi3 r) \\ &=(\tfrac{14}3 \pi +4\sqrt{3})r ,\\ r&=\frac{b}{\tfrac{14}3 \pi +4\sqrt{3}} . \end{align}

  • $\begingroup$ Mind sharing what software did you used to sketch the graphics and label the angles and the lenghts?. Plus, how can be proved that the $PO_{1}O_{2}$ angle formed on the bigger wheel and $BO_{2}$ in the smaller one are the same?. $\endgroup$ – Chris Steinbeck Bell Oct 4 '17 at 2:56
  • $\begingroup$ @ChrisSteinbeckBell The software appears to be $\LaTeX$ $\endgroup$ – gen-z ready to perish Oct 4 '17 at 2:57
  • $\begingroup$ @ChaseRyanTaylor I am not very fond of Latex use in geometry. Do you happen to know any documentation where can I learn how geometry is supported on Latex?. Thanks in advance. $\endgroup$ – Chris Steinbeck Bell Oct 4 '17 at 3:00
  • $\begingroup$ @ChrisSteinbeckBell Share$\LaTeX$ $\endgroup$ – gen-z ready to perish Oct 4 '17 at 3:02
  • $\begingroup$ @ChaseRyanTaylor The documentation provided does not seem to give insight on how to draw euclidean geometry, however there is this site latexdraw.sourceforge.net which does seem to support graphics but I'm not sure if its what the answerer has used in his answer. $\endgroup$ – Chris Steinbeck Bell Oct 4 '17 at 3:10

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