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?