Two rectangular blocks, having identical dimensions, can be arranged either in configuration I or in configuration II as shown in the figure. One of the blocks has thermal conductivity K and the other 2K. The temperature difference between the ends along the x-axis is the same in both configurations. It takes 9s to transport a certain amount of heat from the hot end to the cold end in configuration I. The time to transport the same amount of heat in configuration II is?

Let L be the length of each block. In configuration I, the equivalent thermal resistance is:
R1 = L/K + L/(2K) = (3L)/(2K)
In configuration II, the equivalent thermal resistance is:
1/R2 = 1/(L/K) + 1/(L/(2K)) = 3K/(2L)
R2 = (2L)/(3K)
Since the temperature difference is the same in both configurations, the ratio of heat transported is inversely proportional to the resistance. The time taken to transport the same amount of heat is directly proportional to the resistance.
Therefore, t1/t2 = R1/R2
9/t2 = [(3L)/(2K)]/[(2L)/(3K)] = 9/4
t2 = 4s
However, 4s is not among the options. Let's re-examine the calculation.
In configuration I, the thermal resistance is R1 = L/K + L/2K = 3L/2K
In configuration II, the blocks are in parallel. The equivalent thermal resistance is:
1/R2 = K/L + 2K/L = 3K/L
R2 = L/3K
The heat flow is given by Q/t = ΔT/R, where Q is the amount of heat, t is the time, ΔT is the temperature difference, and R is the thermal resistance.
For configuration I, Q/9 = ΔT/(3L/2K)
For configuration II, Q/t2 = ΔT/(L/3K)
Since Q and ΔT are the same in both configurations, we can write:
(Q/9)/(Q/t2) = [ΔT/(3L/2K)]/[ΔT/(L/3K)]
9/t2 = (2K/3L)*(L/3K) = 2/9
t2 = 4.5s