I just wanted to check to see if I answered this question correctly :) any help would be much appreciated
Q. Air is pumped into a spherical balloon such that the volume of the balloon is increasing at a rate of 3cm^2cm/sec.
(i)Find the rate of change of the radius of the balloon , when the radius is 7cm.Leave your answer in terms of PI.
(ii)Find the rate of change of the surface area of the balloon when the radius is 9cm
What I did:
(i) I differentiated the volume of a sphere to get dv/dt= 4πr^2 . I then subbed 7 in for r and the answer I got was 196π.
(ii) I differentiated the surface area of a sphere which gave me an answer of 8πr.I then subbed in 9 for r and I got 72π as my answer.
We are asked for the rate of change of the radius, that is for $\frac{dr}{dt}$. We have $$3=\frac{dV}{dt}=4\pi r^2\frac{dr}{dt}$$ using $$\frac{dV}{dt}=\frac{dV}{dr}\frac{dr}{dt}$$
Can you finish it now?