Circle Expansion on Expanding Sphere

[Kryptonite-19]

Homework Statement

How would a circle on a sphere expand as a function of the sphere's radius as the sphere expands?

Homework Equations

none were provided

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.

 

[mighty2000]

As the sphere expands, both the radius of the sphere (R) and the radius of the circle on the sphere (r) will increase. Therefore, the rate of change of the circle's radius (dr/dR) will depend on the rate of change of the sphere's radius (dR) and the rate of change of the circle's area (dA). This can be represented by the equation dr/dR = (1/2)(dA/dR)(1/πr). This means that as the sphere expands, the radius of the circle on the sphere will also increase, but at a slower rate compared to the sphere's radius. This is because the circle's area is increasing at a slower rate than the surface area of the sphere.