Resistance of a Wire at varying temperatures

[Enharmonics]

Homework Statement

Determine the resistance of wire at room temperature and at the melting point of the given material.

GIVEN:
Material = Aluminum
Diameter of wire $= 0.5 mm = 5.0 * 10^{-4} m$
Length of wire $= 0.5 m = 5.0 * 10^{-1} m$
Temperature Coefficient: $\alpha = 3.9 * 10^{-3}C^{-1}$
Base Resistivity: $\rho_{0} = 2.82 * 10^{-8} \Omega * m$
Melting Point of Aluminum: $T = 660.3^{\circ} C$
Room Temperature: $T_{0} = 20^{\circ} C$

Homework Equations

Variation of Resistivity w/ Temperature: $\rho = \rho_{0}(1 + \alpha(\Delta T)$

Pouillet's Law: $R = \rho * \frac{L}{A}$

The Attempt at a Solution

I've already finished working the problem out, but in doing so, I had to make a few assumptions I'm not sure are correct. Actually, I'm not even sure if my methodology is correct to begin with.

So, for the Resistance at room temperature, I assumed that the Resistivity given for Aluminum in my textbook is its resistivity at room temperature (not sure if that's right).

The second assumption I made is that the cross-sectional area A of a wire would be given by

$$ A = \pi r^2 = \pi *(\frac{5.0 * 10^{-4} m}{2})^{2}) = 2.0 * 10^{-7} m^{2}$$

Which is based purely on me picturing what each "slice" would look like if I were to cut across a wire an arbitrary number of times. Again, not sure if this is valid, but a circle seemed like the closest geometric shape to what I pictured.

If that's the case, then applying Pouillet's Law yields

$$ R_{room} = \rho_{0} * L/A = (2.82 * 10^{-8} \Omega * m)(\frac{5.0 * 10^{-1} m}{2.0 * 10^{-7} m^{2}}) = 7.1 * 10^{-2} \Omega$$

Then calculating the resistivity of the wire at the melting point I've got

$$ \rho_{mp} = (2.82 * 10^{-8} \Omega * m) + (2.82 * 10^{-8} \Omega * m)(3.9 * 10^{-3}C^{-1})(660.3^{\circ} C - 20^{\circ} C) = 9.9 * 10{-8} \Omega * m$$

So the Resistance of the wire at its melting point is

$$R_{mp} = \rho_{mp} * L/A = (2.82 * 10^{-8} \Omega * m)(\frac{5.0 * 10^{-1} m}{2.0 * 10^{-7} m^{2}}) = 2.5 * 10^{-1} \Omega$$

But is this correct? I can't help but feel like the Resistance of the wire should be higher at the melting point than at room temperature for some reason.

 

[haruspex]

In your last equation line you seem to have copied in the wrong resistivity (ρ₀ instead of ρ_mp), but I take it that's just a typo.
The final answer looks reasonable (it is more than the room temperature resistance!)