Could someone please provide a solution to part A of a rates of change question

Question

Rates of change question from a mathematics textbook about a dissolving pill.

Having a hard time understanding the working and thought process of part (a) of the question. Any insight would be greatly appreciated!

Link: https://i.stack.imgur.com/lxJpK.png

Answer 1

You just need to differentiate the volume $V(t)$ wrt. $t$ and compare with the formula for the surface:

• $V(t)=\frac{4}{3}\pi r^3(t) \Rightarrow \frac{dV}{dt}= 4\pi r^2(t)\cdot \dot r(t)$
• $S(t) = 4\pi r^2(t) \stackrel{\frac{dV}{dt}=kS(t)}{\Longrightarrow}$ $$k\cdot 4\pi r^2(t) = 4\pi r^2(t)\cdot \dot r(t)\Leftrightarrow \boxed{\dot r(t) =k}$$