Develop the model equations for a hot$-$water heater. Consider that the hot$-$water heater is well mixed and has a constant water holdup of $30$ gallons. Assume that initially the water temperature is $120$F$.$ At time equal to zero, hot water is withdrawn at rate of $5$ gallons per minute and $5$ gallons per minute of cold water at $60$F are simultaneously added. Assume that a constant heat addition rate to the hot$-$water heater of $6\backslash$times $104$ BTU$/$h is applied $0\mathrm{.}5$ min after the hot water is removed from the tank. Neglect actuator dynamics. The temperature sensor has a time constant of $0\mathrm{.}25$ min. Solve the problem using Python.

a$.$ What are the dependent variables?

b$.$ What are the independent variables?

c$.$ What are the parameters?

d$.$ Find the degrees of freedom.

e$.$ What are the assumptions in the modeling?

f$.$ How long does it take the tank to reach $90$F if the heating rate is zero?