Answer
(a) $\frac{\rho_2}{\rho_1} = 1$
(b) $\frac{R_2}{R_1} = \frac{1}{2}$
Work Step by Step
(a) The resistivity is a characteristic of the material. Since the material is the same in both wires, the resistivity is the same in both wires.
$\frac{\rho_2}{\rho_1} = 1$
(b) We can write an expression for $R_1$:
$R_1 = \frac{\rho_1~L_1}{A_1}$
$R_1 = \frac{\rho_1~L_1}{\pi~r_1^2}$
We can write an expression for $R_2$:
$R_2 = \frac{\rho_2~L_2}{A_2}$
$R_2 = \frac{\rho_2~L_2}{\pi~r_2^2}$
$R_2 = \frac{\rho_1~(2L_1)}{\pi~(2r_1)^2}$
$R_2 = \frac{1}{2} \times \frac{\rho_1~L_1}{\pi~r_1^2}$
$R_2 = \frac{1}{2} \times R_1$
Therefore, $\frac{R_2}{R_1} = \frac{1}{2}$
