Extending Differentiation (Confused...)

Question

VOLUME : A spherical balloon is inflated with gas at the rate of 500 cubic centimeters per minute. How fast is the radius of the balloon increasing at the instant the radius is (a) 30 cm and (b) 60 cm? Leave answer in terms of pi 

 

 

 

Answer 1

For this kind of problem the hardest part is figuring out what equation you need to use. Look at the various pieces of information and think for a moment what kind of equation has all of that information:

   "VOLUME" <-- the big hint

   "radius"

 

Don't read any further until you have had a minute to think about this.

 

 

 

 

The equation you want is the volume of a sphere:

   V = 4/3*π*r³

 

Since all the information concerns RATES, we will need to take a derivative (with respect to TIME):

   d/dt (V = 4/3*π*r³)

   dV/dt = 4/3*π*(3r²*dr/dt)

   dV/dt = 4πr²*dr/dt

 

I will solve part (a) and leave you to solve part (b). Everything up to this point is exactly the same. From here, just substitute 60 in for 30:

 

We are given dV/dt = 500 cm³/min and r = 30 cm. Substitute these into our derivative equation to get:

   500 cm³/min = 4π(30 cm)²*dr/dt

   500 cm3/min = (3600π cm²)*dr/dt

   5/(36π) cm/min = dr/dt

 

Good luck with part (b). I hope this has been of help.