A $500W$ iron has a base plate of thickness $L_p=0\mathrm{.}5cm,$ made of aluminum alloy $)=3000\frac{kg}{m^3}$;$c_p=(900\frac{J}{kg*K}$ with a high thermal conductivity $)=(180\frac{W}{m*K}.$ One side of the base plate is in contact with resistance wires that generate heat; the other side of the base plate is exposed to air and has a surface area $A_p=0\mathrm{.}03m^2.$ The convective heat transfer coefficient between the base plate and the air is $h=12\frac{W}{m^2*K}.$ Initially, the iron has the same temperature as the surrounding air, $T_{\text{a}\text{i}\text{r}}=20C.$

A$)$ Derive the differential equation that describes the rate of change of the plate temperature, $d\frac{T}{d}t.$

B$)$ Assuming that the resistance wires are delivering the maximum power for the iron and that $90\%$ of the heat generated is transferred to the plate, how long does it take for the plate to reach $140C?$

Normally, the temperature of an iron is maintained at the desired level by switching the electric current through the resistance wires on and off based on a thermostat that senses when the iron reaches the set temperature or gets too low. In our case, the control system is broken and so full power is continuously delivered to the resistance wires.

C$)$ Calculate the equilibrium temperature that the plate of the defective iron would reach at steady state if it is left on$.$ Ignore the effects of radiation.

D$)$ Was it reasonable to ignore radiation in the previous part? Support your answer with a simple calculation.