Stewart sing var calculus intern version page 230 #66

[JrBunny]

I am having some trouble with this proble any help would be appreciated.

Air is being pumped into a spherical weather balloon. At any time t, the volume of
the balloon V(t) and its radius is r(t).
a) What do the derivatives dV/dr and dV/dt represent?
b) Express dV/dt in terms of dr/dt.

The units would be cm^3 but I am still having trouble understanding the problem,
especially a). If you are smart enough to help please do.
Thanks.

 

[galactus]

You're missing the whole idea of differentiating then. dV/dt is the change in volume with regard to the change in radius. As the volume increases, so does the radius. dV/dt is the change in volume as times changes.
If the balloon is being blown up, its volume increases as time passes.

It's a related rates problem. The idea is to find the rate at which one quantity is changing by relating it to other quantities whose rates of change are known.

For part b: differentiate the volume of a sphere formula with respect to r.

 

[JrBunny]

OHHHHHHHHHH!!!!
Of course it all makes sense now!
Thank you very much, you are truly a math wizard.