Linearization estimating Tolerance

[atowriss]

I am stuck on this probelm, and any help would be greatrly appreciated

How accurately must the interior of a 10m high cylindrical tank be measured to calculate the tank's volume within 1% of it's true vale?

So, I know that V=10pir^2, and that the estimated volume (V') needs to be no greater than Actual Volume (V) by 1%.

Taking the derivative of the above, I get V ' = 20pirr' (r' is r prime), and thus v'/v = 2r'/r. So, 2r'/r has to be less than or equal to .01.
What I can't figure out is how to get the change in r (dr) so this is true.

If I was given the change in radius, I know how to estimate the volume, and compare the 2, but I seem to be at a loss as to how to determine what the change in radius is.

Thanks so much

 

[tkhunny]

You have it. Just put it all together.

\(\displaystyle dV < 0.01 \cdot V \implies 20 \cdot \pi \cdot r \cdot dr < 0.01 \cdot 10 \cdot \pi r^{2} \implies dr < \frac{r}{200}\)

Okay, no we're done with the algebra. What does it mean?