Question Number 127749 by ajfour last updated on 01/Jan/21
Commented by ajfour last updated on 01/Jan/21
$${Find}\:{ratio}\:{of}\:{radius}\:{of}\:{small} \\ $$$${semicircles}\:{of}\:{equal}\:{radii}\:{to} \\ $$$${that}\:{of}\:{outer}\:{semicircle}\:{r}/{R}. \\ $$
Answered by mr W last updated on 01/Jan/21
$${let}\:\lambda=\frac{{r}}{{R}} \\ $$$$\frac{{R}−{r}}{{R}}=\frac{{r}}{\mathrm{2}{R}−{r}} \\ $$$$\mathrm{1}−\lambda=\frac{\lambda}{\mathrm{2}−\lambda} \\ $$$$\lambda^{\mathrm{2}} −\mathrm{4}\lambda+\mathrm{2}=\mathrm{0} \\ $$$$\Rightarrow\lambda=\mathrm{2}−\sqrt{\mathrm{2}} \\ $$
Commented by mr W last updated on 01/Jan/21
Commented by ajfour last updated on 01/Jan/21
$${Excellent}\:{notice},\:{Sir};\:{what}\:{to}\:{say}... \\ $$$$\mathcal{A}\:\mathcal{H}{app}\mathcal{Y}\:\mathcal{N}{ew}\:\mathcal{Y}{ear}\:!\: \\ $$
Commented by mr W last updated on 01/Jan/21
$${thanks}\:{alot}! \\ $$$${happy}\:{new}\:{year}\:{to}\:{you}\:{too}! \\ $$