# Circle Expansion on Expanding Sphere

1. ### Kryptonite-19

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

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