Answer
The ratio of the diameter of the copper wire to that of the aluminum wire is 0.80
Work Step by Step
We can find an expression for the resistance of the copper wire:
$R = \frac{\rho_c~L}{A_c}$
$R = \frac{\rho_c~L}{\pi~(\frac{d_c}{2})^2}$
$R = \frac{4~\rho_c~L}{\pi~d_c^2}$
We can find an expression for the resistance of the aluminum wire:
$R = \frac{\rho_a~L}{A_a}$
$R = \frac{\rho_a~L}{\pi~(\frac{d_a}{2})^2}$
$R = \frac{4~\rho_a~L}{\pi~d_a^2}$
Since the resistance is equal in both wires, we can equate the two expressions:
$\frac{4~\rho_a~L}{\pi~d_a^2} = \frac{4~\rho_c~L}{\pi~d_c^2}$
$\frac{d_c^2}{d_a^2} = \frac{\rho_c}{\rho_a}$
$\frac{d_c}{d_a} = \sqrt{\frac{\rho_c}{\rho_a}}$
$\frac{d_c}{d_a} = \sqrt{\frac{1.68\times 10^{-8}~\Omega~m}{2.65\times 10^{-8}~\Omega~m}}$
$\frac{d_c}{d_a} = 0.80$
The ratio of the diameter of the copper wire to that of the aluminum wire is 0.80
