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Question Number 213323 by Spillover last updated on 02/Nov/24

Commented by Frix last updated on 03/Nov/24

$No nice numbers$ $Let r = 1$ $\Rightarrow$ $ρ_{q . c .} \approx 4.87164313433 [_{polynome of 8^{th} degree}^{Solution of a}]$ $ρ_{s . c .} \approx 3.49740513991$ $R \approx 1.66629777247$
	Answered by mr W last updated on 03/Nov/24

	Commented by mr W last updated on 03/Nov/24

	$a = r a d i u s o f q u a r t e r c i r c l e$ $b = r a d i u s o f s e m i c i r c l e$ ${(a - r)}^{2} - r^{2} = {(b + r)}^{2} - {(b - r)}^{2}$ $\Rightarrow a^{2} - 2 a r = 4 b r . . . (i)$ $\sqrt{{(a - R)}^{2} - R^{2}} + \sqrt{{(b - R)}^{2} - R^{2}} = b$ $\Rightarrow \sqrt{a^{2} - 2 a R} + \sqrt{b^{2} - 2 b R} = b . . . (i i)$ $\sqrt{{(a + r)}^{2} - r^{2}} = b + \sqrt{{(b - r)}^{2} - r^{2}}$ $\Rightarrow \sqrt{a^{2} + 2 a r} = b + \sqrt{b^{2} - 2 b r} . . . (i i i)$ $l e t α = \frac{a}{r}, β = \frac{b}{r}, λ = \frac{R}{r}$ $α^{2} - 2 α = 4 β . . . (i)$ $\sqrt{α^{2} - 2 α λ} + \sqrt{β^{2} - 2 β λ} = β . . . (i i)$ $\sqrt{α^{2} + 2 α} = β + \sqrt{β^{2} - 2 β} . . . (i i i)$ $\Rightarrow λ \approx 1.666297772$
	Commented by Spillover last updated on 03/Nov/24

	$p e r f e c t$
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