A drill is being used to drill a hole in a thick metal plate. Assume the variation of temperature, T^oC from some background value of the drill bit along its length, x (cm), x ? [0, 10] and at time t (sec), t ? [0, ?] is T(x, t). Friction at the active end of the drill leads to a temperature T(10, t) = 400(1 ? e^-t ). Heat is lost through the side of the drill at a rate ? = 0.01, and it is insulated at x = 0 so that the full system becomes