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Question Number 59121 by naka3546 last updated on 05/May/19

	Answered by mr W last updated on 05/May/19

	$c i r v l e 1 :$ $r_{1} = 1$ $a_{1} = s q u a r e s i d e l e n g t h$ $a_{1}^{2} + {(\frac{a_{1}}{2})}^{2} = r_{1}^{2}$ $\Rightarrow a_{1} = \frac{2}{\sqrt{3}} r_{1} = \frac{2 \sqrt{3}}{3} r_{1}$ $s_{1} = l e n g t h o f r e d p a t h$ $s_{1} = 3 a_{1} + r_{1} - \frac{a_{1}}{2} = (2 \sqrt{3} + 1 - \frac{\sqrt{3}}{3}) r_{1} = (\frac{5 \sqrt{3}}{3} + 1) r_{1} = {k r}_{1}$ $w i t h k = \frac{5 \sqrt{3}}{3} + 1 \approx 3.887$ $c i r c l e 2 :$ $r_{2} = \frac{a_{1}}{2} = \frac{\sqrt{3}}{3} r_{1}$ $s_{2} = {k r}_{2} = \frac{\sqrt{3}}{3} {k r}_{1} = {q s}_{1}$ $w i t h q = \frac{\sqrt{3}}{3} < 1$ $t o t a l l e n g t h o f r e d p a t h = S$ $S = s_{1} + s_{2} + . . . + s_{n} + . . . \leftarrow G . P .$ $= \frac{s_{1}}{1 - q} = \frac{{k r}_{1}}{1 - \frac{\sqrt{3}}{3}} = \frac{3}{3 - \sqrt{3}} \times \frac{5 \sqrt{3} + 3}{3} \times 1$ $= \frac{(5 \sqrt{3} + 3) (3 + \sqrt{3})}{6}$ $= 4 + 3 \sqrt{3}$
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