The outer surface of an engine is located in an area where oil leakage may occur. Some oils have autoignition temperatures around $250C.$ If oil contacts a hot engine surface that exceeds its autoignition temperature, spontaneous ignition can occur. Consider the engine housing as a plane wall, with the inner surface $)$ subjected to a heat flux of $6k\frac{W}{m^2}.$ The engine housing, with a thermal conductivity $13\mathrm{.}5\frac{W}{m}*K,$ is $1$ cm thick, and the outer surface $(x=L)$ is exposed to ambient air at $35C$ with a convection heat transfer coefficient of $20\frac{W}{m^2}*K.$ To prevent fire risk if leaked oil contacts the engine's hot surface, the temperature of the engine surface must be kept below $200C.$ Determine the temperature profile within the engine housing and the temperatures of both the inner and outer surfaces. Also, assess whether the outer surface temperature is below the oil's autoignition temperature.