# 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.

Result

n =  4

#### Solution:

$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=4 \ \\ \ \\ θ=180/n=180/4=45 \ ^\circ \ \\ ρ=\dfrac{ r_{1} \cdot \ \sin(θ) }{ 1-\sin(θ) }=\dfrac{ 1 \cdot \ \sin(45^\circ ) }{ 1-\sin(45^\circ ) } \doteq 2.4142 \ \\ R_{2}=r + 2 \cdot \ ρ=4 + 2 \cdot \ 2.4142 \doteq 8.8284 \ \\ R_{2}

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