Q$4.$ In a food processing facility, a spherical container of inner radius $r_1=39cm,$ outer radius $r_2$

$=40\mathrm{.}5cm,$ and thermal conductivity $k=1\mathrm{.}5\frac{W}{m}.C$ is used to store hot water and to keep it at

$105C$ at all times. To accomplish this, the outer surface of the container is wrapped with a $510$

W electric strip heater and then insulated. The temperature of the inner surface of the container

is observed to be nearly $105C$ at all times. Assuming $15\%$ of the heat generated in the heater is

lost through the insulation, $($a$)$ express the differential equation and the boundary conditions for

steady one$-$dimensional heat conduction through the container, $($b$)$ obtain a relation for the

variation of temperature in the container material by solving the differential equation, and $($c$)$

evaluate the outer surface temperature of the container. Also determine how much water at $100C$

this tank can supply steadily if the cold water enters at $20C.$

$($Note: The specific heat of water at its average temperature can be found directly from Table A$-$

Work in SI units and the final result should be in SI units$)$