Answer
We can rank the rods according to their coefficients of thermal expansion:
$c \gt a = b = d$
Work Step by Step
We can write an expression for the coefficient of thermal expansion:
$\Delta L = L~\alpha~\Delta T$
$\alpha = \frac{\Delta L}{L~\Delta T}$
We can find the coefficient of thermal expansion for each rod:
Rod a:
$\alpha = \frac{4\times 10^{-4}~m}{(2~m)(10~C^{\circ})} = 20\times 10^{-6}~/C^{\circ}$
Rod b:
$\alpha = \frac{4\times 10^{-4}~m}{(1~m)(20~C^{\circ})} = 20\times 10^{-6}~/C^{\circ}$
Rod c:
$\alpha = \frac{8\times 10^{-4}~m}{(2~m)(10~C^{\circ})} = 40\times 10^{-6}~/C^{\circ}$
Rod d:
$\alpha = \frac{4\times 10^{-4}~m}{(4~m)(5~C^{\circ})} = 20\times 10^{-6}~/C^{\circ}$
We can rank the rods according to their coefficients of thermal expansion:
$c \gt a = b = d$
