Question Number 223544 by ajfour last updated on 29/Jul/25
Commented by ajfour last updated on 29/Jul/25
$${If}\:\mathrm{2}{R}\theta=\mathrm{2}\pi{r},\:{find}\:{R}/{r}\:{for}\:{h}=\mathrm{0}. \\ $$
Answered by mr W last updated on 29/Jul/25
$$\theta{R}=\pi{r} \\ $$$$\left({R}+{r}\right)\mathrm{cos}\:\theta=\left({R}+{h}−{r}\right) \\ $$$$\left(\mathrm{1}+\frac{{r}}{{R}}\right)\mathrm{cos}\:\frac{\pi{r}}{{R}}=\left(\mathrm{1}+\frac{{h}}{{R}}−\frac{{r}}{{R}}\right) \\ $$$${with}\:{h}=\mathrm{0}: \\ $$$$\Rightarrow\mathrm{cos}\:\frac{\pi{r}}{{R}}=\frac{\mathrm{1}−\frac{{r}}{{R}}}{\mathrm{1}+\frac{{r}}{{R}}}\:\Rightarrow\frac{{r}}{{R}}=\frac{\mathrm{1}}{\mathrm{3}} \\ $$
Commented by ajfour last updated on 29/Jul/25
Commented by mr W last updated on 29/Jul/25
$$\mathrm{2}\theta{R}=\mathrm{2}\pi{r}\:{doesn}'{t}\:{mean}: \\ $$
Commented by mr W last updated on 29/Jul/25
Commented by ajfour last updated on 29/Jul/25
$${yes}\:{sir},\:{i}\:{cant}\:{really}\:{impose} \\ $$$${further}\:{condition}\:\pi{r}={R}\theta \\ $$$${Thanks}! \\ $$