Circle Expansion on Expanding Sphere

  1. 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

    [tex]\S=4\,\pi \,{R}^{2}[/tex]
    [tex]A=\pi \,{r}^{2}[/tex]

    [tex]{\it dA}=2\,\pi \,r{\it dr}[/tex]
    [tex]{\it dS}=8\,\pi \,R{\it dR}[/tex]

    [tex]{\frac {{\it dA}}{{\it dR}}}=2\,{\frac {\pi \,r{\it dr}}{{\it dR}}}[/tex]

    [tex]{\frac {{\it dA}}{{\it dS}}}{\frac {{\it dS}}{{\it dR}}}={\frac {{\it dA}}{{\it dR}}}[/tex]

    [tex]{\frac {{\it dr}}{{\it dR}}}=1/2\,{\frac {{\it dA}}{{\it dR}\,\pi \,r}}[/tex]


    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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