Problem $6$

Consider an iron whose base plate is made of an aluminum alloy of thickness $\backslash ($ L $\backslash ),$ density $\backslash (\backslash$rho $\backslash ),$ specific heat capacity $\backslash ($ c$\_\{$p$\}\backslash ),$ and thermal diffusivity of $\backslash (\backslash$alpha $\backslash ).$ The base plate has a surface area of $\backslash ($ A $\backslash )$ and the electrical power consumption of the iron is $\backslash ($ P $\backslash ).$ Initially the iron is in thermal equilibrium with the ambient air at $\backslash ($ T$\_\{\backslash$infty$\}\backslash ).$ Taking the heat transfer coefficient of the base plate to be $\backslash ($ h $\backslash )$ and assuming $85$ percent of the heat generated in the resistance wires is transferred to the plate, what would be the final, steady state temperature $\backslash ($ T$\_\{0\}\backslash )$ that the iron would reach? Derive an expression for the rate of heat loss from the iron as a function of time. How long should a user wait before using the iron so that it's at fully operational temperature? Under what condition is it realistic to assume the plate temperature to be uniform at all times?