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Question Number 191128 by Mingma last updated on 18/Apr/23

	Answered by mr W last updated on 18/Apr/23

	Commented by mr W last updated on 19/Apr/23

	$b = \sqrt{{(a + r)}^{2} - r^{2}} = \sqrt{a (a + 2 r)}$ $\frac{b}{a + r} = \frac{a + 2 r}{b + a}$ $b (b + a) = (a + r) (a + 2 r)$ $a^{3} - r^{2} a - 2 r^{3} = 0$ $l e t λ = \frac{a}{r}$ $λ^{3} - λ - 2 = 0$ $\Rightarrow λ = \sqrt[3]{\frac{\sqrt{78}}{9} + 1} - \sqrt[3]{\frac{\sqrt{78}}{9} - 1}$ $R = \frac{a + b}{2}$ $\frac{r}{R} = \frac{2 r}{a + b} = \frac{2}{λ + \sqrt{λ (λ + 2)}} \approx 0.52138$
	Commented by Mingma last updated on 19/Apr/23
	please check your solution well. the final answer should be 0.27
	Commented by mr W last updated on 19/Apr/23
	$i have expected more care from you. are you really only interested in the final value, not what the value means and how the value was obtained? i got (r/R)≈0.52138 and i didn′t say that this is the fraction of the shaded area. the fraction of shaded area is certainly ((r/R))^2 ≈0.2718.$
	$i h a v e e x p e c t e d m o r e c a r e f r o m y o u .$ $a r e y o u r e a l l y o n l y i n t e r e s t e d i n t h e$ $f i n a l v a l u e, n o t w h a t t h e v a l u e m e a n s$ $a n d h o w t h e v a l u e w a s o b t a i n e d ?$ $i g o t \frac{r}{R} \approx 0.52138 a n d i {d i d n}^{'} t s a y t h a t$ $t h i s i s t h e f r a c t i o n o f t h e s h a d e d$ $a r e a . t h e f r a c t i o n o f s h a d e d a r e a$ $i s c e r t a i n l y {(\frac{r}{R})}^{2} \approx 0.2718 .$
	Commented by Mingma last updated on 19/Apr/23
	Clearly, you're right! Thank you, sir!
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