Expand the quadratic term for resistance equation

[amcavoy]

A cylindrical wire of length \ell and cross-sectional area \textrm{A} has a fixed volume \textrm{V}. If \ell is increased by +x percent and the volume and resistivity stay the same, by what percentage (in terms of x) will the resistance change?

I said that A\ell=A_{\textrm{new}}\ell\left(1+\frac{x}{100}\right)\implies A_{\textrm{new}}=\frac{A}{1+\frac{x}{100}}. Also, we know that R\propto\frac{\ell}{A}.. Therefore,

R_{\textrm{old}}\propto\frac{\ell}{A}

R_{\textrm{new}}\propto\frac{\ell\left(1+\frac{x}{100}\right)^{2}}{A}

Somehow I'm not getting as nice of an answer as I expected.

 

[Integral]

Expand the quadradic term, you can claim that the {(\frac x {100})}^2 term is very small and ignore it. Can you see where to go from there?

 

[amcavoy]

So roughly 2x percent?