Related rates & unknown factors

[Michael Santos]

Homework Statement

Your blowing up a balloon at a rate of 300 cubic inches per minute. When the balloon's radius is 3 inches, how fast is the radius increasing?

Homework Equations

The Attempt at a Solution

I know the answer to this question. It is approximately 2.65 inches per minute, what my question is; the area is 113.0973355 when the radius equals 3. when 300 and 2.65 are divided by 60 sec you find that the area is increasing at 5 inches per sec and the radius is increasing at 0.04416666667 per sec, tho when you input 3.04416666667 in the formula for a sphere the area comes out to be 118.1663682 just over 5 inches, what is this unknown factor that participates in the growth of the area or is this a physics problem, this continues to happen if you divide even further. It is the same thing for when you increase the radius of 3 by 2.65 and find the area which says it would be 300 cubic inches wider as the area increases at 300 cubic inches per minute but it is not.

 

[Dewgale]

You're making a mistake in your assumptions. Specifically, you're assuming that just because the volume changes constantly with time, so too must the radius. You can see by a very quick application of the chain rule that this isn't true, but you can also think about it physically -- If you want to add 300 cubic inches of volume to a balloon, that will make the radius change a lot more when the balloon is empty than it will if the balloon is a mile wide.

 

[Michael Santos]

You're making a mistake in your assumptions. Specifically, you're assuming that just because the volume changes constantly with time, so too must the radius. You can see by a very quick application of the chain rule that this isn't true, but you can also think about it physically -- If you want to add 300 cubic inches of volume to a balloon, that will make the radius change a lot more when the balloon is empty than it will if the balloon is a mile wide.

Yes i see

 

[RPinPA]

I assume this is a calculus class (correct me if I'm wrong). So the principle here is the Chain Rule.
$\frac {dV} {dt} = \frac {dV}{dr} \frac {dr} {dt}$

You obtain the expression for $\frac {dV}{dr}$ by differentiating the formula for volume of a sphere with respect to the radius.
The rate you were given is $\frac {dV}{dt}$. And you're being asked for $\frac {dr}{dt}$, so you can easily solve for that by plugging in the other information.