Q$.5$ A heat generating slab A $($thickness $=0\mathrm{.}25m,k_A=15\frac{W}{m}C)$ is sandwiched between two other slabs $B($thickness $=0\mathrm{.}1m,k_B=10\frac{W}{m}C)$ and $C($thickness $=0\mathrm{.}15m,kg=30\frac{W}{m}C)$ as shown in fig. There is no heat generation in slab $B$ or $C.$ The temperature distribution in slab $A$ is known to be $T_A=90+4500x-11000x^2,$ where $T$ is in $C$ and $x$ is the distance $($in meters$)$ from the left surface of $B$ as shown in fig. The wall $B$ is in contact with a fluid at temperature $T_1=40C,$ the wall heat transfer coefficient being $h_1.$ Similarly, the free surface of $C$ loses heat to a medium at a temperature $35C,$ and the heat transfer coefficient being $h_2$ Assume steady state condition.

$($a$)$ Calculate the temperatures at the surface of slab A$.$ What is the maximum temperature in A and where does it occur?

$($b$)$ Determine the temperature gradients at both the surfaces of each of the slabs A$,$ B$,$ and C$.$ Comment on the signs of the gradients.

$($c$)$ Find the temperature profiles in slabs B and C$.$ Also, calculate the values of the heat transfer coefficient $h_1$ and $h_2.$

$6$