Circle Expansion on Expanding Sphere

by Kryptonite-19
Tags: circle, expanding, expansion, sphere
 P: 5 1. The problem statement, all variables and given/known data How would a circle on a sphere expand as a function of the sphere's radius as the sphere expands? 2. Relevant equations none were provided 3. The attempt at a solution $$\S=4\,\pi \,{R}^{2}$$ $$A=\pi \,{r}^{2}$$ $${\it dA}=2\,\pi \,r{\it dr}$$ $${\it dS}=8\,\pi \,R{\it dR}$$ $${\frac {{\it dA}}{{\it dR}}}=2\,{\frac {\pi \,r{\it dr}}{{\it dR}}}$$ $${\frac {{\it dA}}{{\it dS}}}{\frac {{\it dS}}{{\it dR}}}={\frac {{\it dA}}{{\it dR}}}$$ $${\frac {{\it dr}}{{\it dR}}}=1/2\,{\frac {{\it dA}}{{\it dR}\,\pi \,r}}$$ Stuck at this point. 1. The problem statement, all variables and given/known data 2. Relevant equations 3. The attempt at a solution