Question

The area of a circular sun spot is growing at a rate of 1,200 km2/s.

Answer 1

A)

$\bf \textit{area of a circle}\\\\ A=\pi r^2 \\\\\\ \cfrac{dA}{dt}=\pi \cdot 2r\cfrac{dr}{dt}\implies \cfrac{dA}{dt}=2\pi r\cfrac{dr}{dt}\implies \cfrac{\frac{dA}{dt}}{2\pi r}=\cfrac{dr}{dt}\\\\ -----------------------------\\\\ \left. \cfrac{dr}{dt} \right|_{r=3000}\implies \cfrac{1200}{2\pi \cdot 3000}=\cfrac{dr}{dt}$

B)

$\bf A=\pi r^2\qquad A=490000\implies 490000=\pi r^2\implies \sqrt{\cfrac{490000}{\pi }}=r \\\\\\ \left. \cfrac{dr}{dt} \right|_{r=\sqrt{\frac{490000}{\pi }}}\implies \cfrac{1200}{2\pi \cdot \sqrt{\frac{490000}{\pi }}}=\cfrac{dr}{dt}$