$3.(40$ pts $+5$ pts bonus$)$ A wire is made by extruding heated metal through a circular die and then allowing the wire to cool in air prior to being wound on a spool. The arrangement is shown in the figure below. The spool rotates in such a way that the velocity $($v$\_0)$ of the wire in the x direction is constant. The diameter of the wire is D and the length of wire between the heater and the spool is L$.$ The wire leaves the heated die at a temperature of T$,$ and the air temperature is T$\_0.$ The wire material has a density of $\backslash$rho and a heat capacity of C$\_$p$.$ You may assume that the thermal conductivity of the wire is constant and equal to k$,$ and the heat transfer coefficient between the wire and the air, h$,$ is also constant. Further assuming that this process has reached steady state and the temperature of the wire on reaching the spool is T$\_$a$,$ do the following: a$)(10$ pts$)$ When the wire that is not thin, the wire temperature T varies in both x and r directions. List all appropriate assumptions and simplify the equation of energy to obtain the differential equation describing the temperature of the wire, T$.$ b$)(10$ pts$)$ When the wire is thin but very long, list the assumptions different from a$)$ and reduce the equation of energy to obtain the differential equation describing the temperature of the wire, T$.$ c$)(16$ pts$)$ Solve the differential equation obtained in b$)$ with the appropriate boundary conditions to give the temperature distribution along the wire. d$)(4$ pts$)$ Simplify the differential equation in b$)$ for the case when v$\_0=0$ and show the general solution for the differential equation containing integration constants. You do not need to identify the boundary conditions or obtain the final temperature profile of the wire. e$)(5$ pts bonus$)$ When the wire is not moving in a$),$ simplify the obtained differential equation of energy and write down the appropriate boundary conditions that are necessary to obtain the wire temperature profile.