Question

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

(a) How fast is the radius growing at the instant when it equals 5,000 km?

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{(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{(490000)/(\pi )}}\implies \cfrac{1200}{2\pi \cdot \sqrt{(490000)/(\pi )}}=\cfrac{dr}{dt}`