A perfectly mixed, stirred cylindrical tank with a volume capacity of $30m^3$ contains water at $20C$ that occupies one third of its maximum volume. This tank is equipped with a spiral coil$-$type heating element. Heat required for maintaining the water temperature is supplied by a steam flow that travels through the coil, providing a constant heat flux denoted as $q_{in}^{}=600k\frac{J}{m^2}s.$

Heat transfer area of the heating element available to the water is directly proportional to the volume of liquid present in the tank at any given time $t,$ described by the relationship:

$A(t)=\frac{A_{\text{t}\text{o}\text{t}}}{V_{\text{t}\text{o}\text{t}}}V(t)$

where $A_{\text{t}\text{o}\text{t}}$ is the total surface area of the heating coil and $V_{\text{t}\text{o}\text{t}}$ is the total volume of the tank. The total surface area of the heating coil is given as $300m^2.$ At time $t=0,$ water at a temperature of $20C$ begins to be added to the tank at a rate of $1\frac{m^3}{s}.$

Determine the temperature profile of the water inside the tank, $T(t),$ from the initial condition until the tank overflows and calculate the temperature of the water when the tank is completely illed.