Question-127749

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 λ=(r/R) ((R−r)/R)=(r/(2R−r)) 1−λ=(λ/(2−λ)) λ^2 −4λ+2=0 ⇒λ=2−(√2)

$${let}\:\lambda=\frac{{r}}{{R}} \\ $$$$\frac{{R}−{r}}{{R}}=\frac{{r}}{\mathrm{2}{R}−{r}} \\ $$$$\mathrm{1}−\lambda=\frac{\lambda}{\mathrm{2}−\lambda} \\ $$$$\lambda^{\mathrm{2}} −\mathrm{4}\lambda+\mathrm{2}=\mathrm{0} \\ $$$$\Rightarrow\lambda=\mathrm{2}−\sqrt{\mathrm{2}} \\ $$

Commented by mr W last updated on 01/Jan/21