# [SOLVED] error propagation: the approximation formula

• Aug 3rd 2008, 09:51 AM
sinewave85
[SOLVED] error propagation: the approximation formula
I can't seem to get the right answer to this one, and I hope someone can tell me what I am doing wrong!

"You measure the radius of a circle to be 12 cm and use the formula A = pi(r^2) to calculate the area. If your measurement of the radius is accurate to within 3%, approximately how accurate (to the nearest percent) is your calculation of the area?

$A(r) = \pi r^2$
$A'(r) = 2\pi r$
$\left|\frac{dA}{A}\right|= \left|\frac{2 \pi r dr}{ \pi r^2}\right|$
$\left|\frac{dA}{A}\right| = 2\left|\frac{dr}{r}\right|$
$\left|\frac{dA}{A}\right| = 2\left|\frac{\pm0.03(10)}{12}\right|$
$\left|\frac{dA}{A}\right| = 0.05$

The problem is that the answer given in the back of the book is 0.06. So, where did I mess up? Thanks for the help!
• Aug 3rd 2008, 10:12 AM
flyingsquirrel
Hi
Quote:

Originally Posted by sinewave85
I can't seem to get the right answer to this one, and I hope someone can tell me what I am doing wrong!

"You measure the radius of a circle to be 12 cm and use the formula A = pi(r^2) to calculate the area. If your measurement of the radius is accurate to within 3%, approximately how accurate (to the nearest percent) is your calculation of the area?

$A(r) = \pi r^2$
$A'(r) = 2\pi r$
$\left|\frac{dA}{A}\right|= \left|\frac{2 \pi r dr}{ \pi r^2}\right|$
$\left|\frac{dA}{A}\right| = 2\left|\frac{dr}{r}\right|$

Taking it from here, the relative error of $r$ is $\left|\frac{\mathrm{d}r}{r}\right|=0.03$ so $\left|\frac{\mathrm{d}A}{A}\right|=2\times 0.03=0.06$. Be careful to make a difference between absolute error ( $|\mathrm{d}r|$ ) and relative error. $\left( \left|\frac{\mathrm{d}r}{r}\right| \right)$