Question

A spherical balloon is filling with air at a rate of 24 pi cubic inches per minute. How fast is the radius of the balloon increasing when the volume is 288 pi cubic inches?

Answer 1

Are you sure the question is right?

Answer 2

You would need to be given a starting radius to come up with an answer, but you can get this far from that information.

Answer 3

dr/dt= 24 (we will ignore units for now as it is too complicated to type)
Volume formula is 4/3pir^3 which if we derive we get V'=4pir^2. Now we can put 288=4pir^2(24) which would give us r= 0.977 which pretty sure I did it right... I think

Answer 4

24 is the rate of change of the volume, not the radius. so dV/dt = 24.

And to do a related rates problem you need to differentiate implicitly. Your derivative would look like this:
\[V \frac{ dV }{ dt }= 4(pi)r ^{2}\frac{ dr }{ dt }\]