Annulus

Two concentric circles with radii 1 and 9 surround the annular circle. This ring is inscribed with n circles that do not overlap. Determine the highest possible value of n.

Final Answer:

n =  3

Step-by-step explanation:

$$\begin{gathered} r_1=1 \\ {r}_2=9 \\ r=(r_2-r_1)/2=(9-1)/2=4 \\ R=r_1+r=1+4=5 \\ a=r=4 \\ {o}_1=2\pi \cdot R=2\cdot 3.1416\cdot 5\doteq 31.4159 \\ {n}_1=o_1/(2\cdot r)=31.4159/(2\cdot 4)\doteq 3.927 \\ n=3 \\ \theta =180/n=180/3=60{}^{\circ } \\ \rho =\frac{r_1\cdot \sin \theta }{1-\sin \theta }=\frac{r_1\cdot \sin 60°}{1-\sin 60°}=\frac{1\cdot \sin 60°}{1-\sin 60°}=\frac{1\cdot 0.866025}{1-0.866025}=6.4641 \\ {R}_2=r+2\cdot \rho =4+2\cdot 6.4641\doteq 16.9282 \\ {R}_2<R \end{gathered}$$

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