I am confused with this question, difficult to find steps to solve it.

A carbon cable and a tungsten cables are connected each and made a cable. The resistivity of tungsten is 5.6*10^-8Ωm and carbon is 3.5*10^-5Ωm. The temperature coefficient of resistivity ,tungsten=0.0045°C-1 and carbon=-0.0005°C-1. The resistance of resultant cable is not change with the temperature. **What is the ratio of the two lengths of the carbon and tungsten cables.**(Assume that the two cables have same cross sectional area).

Pouillet's law is

\(\displaystyle R = \rho \frac{l}{A}\)

where R is resistance, \(\displaystyle \rho\) is the resistivity, l is the length of the cable and A is the cross-sectional area of the cable. Furthermore, the resistivity can be temperature-dependent, which is usually expressed as a linear relationship relative to some reference resistivity, \(\displaystyle \rho_0\), at a reference temperature, \(\displaystyle T_0\).

\(\displaystyle \rho = \rho_0 \left[1 + \alpha(T-T_0)\right]\)

where T is temperature and \(\displaystyle \alpha\) is the temperature coefficient of resistivity.

Using these laws we can build a relationship for the resistance of the whole cable. We know that two cables attached end to end add up in series, so

\(\displaystyle R = R_c + R_t\)

\(\displaystyle \rho \frac{(l_t + l_c)}{A}= \rho_{0,t}\frac{l_t}{A} \left[1 + \alpha_t(T-T_0)\right] + \rho_{0,c}\frac{l_c}{A} \left[1 + \alpha_c(T-T_0)\right]\)

\(\displaystyle \rho (l_t + l_c)= \rho_{0,t} l_t \left[1 + \alpha_t(T-T_0)\right] + \rho_{0,c} l_c \left[1 + \alpha_c(T-T_0)\right]\)

If we expand the brackets and then rearrange this so that all of the terms not dependent on T are on the LHS and all the ones that are are on the RHS, we get

\(\displaystyle \rho (l_t + l_c) - \rho_{0,t} l_t + \rho_{0,t} l_t \alpha_t T_0 - \rho_{0,c} l_c + \rho_{0,c} l_c \alpha_c T_0 = \rho_{0,t} l_t \alpha_t T + \rho_{0,c} l_c \alpha_c T\)

Since we know there is no temperature dependence, we can say that

\(\displaystyle \rho_{0,t} l_t \alpha_t T + \rho_{0,c} l_c \alpha_c T = 0\)

Which can occur because \(\displaystyle \alpha_c\) is negative. that leaves

\(\displaystyle \rho (l_t + l_c) - \rho_{0,t} l_t + \rho_{0,t} l_t \alpha_t T_0 - \rho_{0,c} l_c + \rho_{0,c} l_c \alpha_c T_0 = 0\)

I think it should be fairly straightforward to rearrange this to get the ratio of the lengths of the cables.