Answer
We can rank the wires in order of decreasing resistance:
$d = e \gt a \gt b = c = f$
Work Step by Step
We can write an expression for the resistance of a wire:
$R = \frac{\rho~L}{A}$
Let the diameter be 1 mm. Then $r_0 = 0.5~mm$
Let the resistivity $\rho_0$ be the resisitivity of aluminum.
Let the length $L_0 = 1~m$
Then: $R_0 = \frac{\rho_0~L_0}{\pi~r_0^2}$
We can find an expression for the resistance of each wire:
(a) $R_a = \frac{(2\rho_0)~(L_0)}{(\pi)~(2r_0)^2} = 0.5~R_0$
(b) $R_b = \frac{(2\rho_0)~(2L_0)}{(\pi)~(4r_0)^2} = 0.25~R_0$
(c) $R_c = \frac{(\rho_0)~(L_0)}{(\pi)~(2r_0)^2} = 0.25~R_0$
(d) $R_d = \frac{(\rho_0)~(L_0)}{(\pi)~(r_0)^2} = R_0$
(e) $R_e = \frac{(2\rho_0)~(2L_0)}{(\pi)~(2r_0)^2} = R_0$
(f) $R_f = \frac{(\rho_0)~(4L_0)}{(\pi)~(4r_0)^2} = 0.25~R_0$
We can rank the wires in order of decreasing resistance:
$d = e \gt a \gt b = c = f$
