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Question Number 186408 by normans last updated on 04/Feb/23

Answered by mr W last updated on 04/Feb/23

$$R=radiusofbigcircle$$$$r=radiusofsmallcircles$$$${(R-r)}^2={\left(3r\right)}^2+r^2$$$$9r^2+2Rr-R^2=0$$$$r=\frac{(\sqrt{10}-1)R}{9}$$$$\sin \frac{\theta }{2}=\frac{r}{R+r}=\frac{1}{\sqrt{10}+2}$$$$\theta =2{\sin }^{-1}\frac{1}{\sqrt{10}+2}$$$$n=\lfloor \frac{2\pi }{\theta }\rfloor =\lfloor \frac{\pi }{{\sin }^{-1}\frac{1}{\sqrt{10}+2}}\rfloor =16$$$$i.e.max.16smallcirclesfitintothe$$$$ring.$$

Commented by normans last updated on 04/Feb/23

$$\mathit{g}\mathit{r}\mathit{e}\mathit{a}\mathit{t}\mathit{S}\mathit{i}\mathit{r},\mathit{Y}\mathit{o}\mathit{u}\mathit{r}\mathit{e}\mathit{a}\mathit{l}\mathit{l}\mathit{y}\mathit{a}\mathit{r}\mathit{e}\mathit{a}\mathit{g}\mathit{e}\mathit{o}\mathit{m}\mathit{e}\mathit{t}\mathit{r}\mathit{y}\mathit{e}\mathit{x}\mathit{p}\mathit{e}\mathit{r}\mathit{t}$$

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