(Roberts7) A city is divided into 3 areas 1, 2, and 3. It is estimated that amounts u1, u2, and u3 of pollution are emitted each day from these three areas. A fraction qij of the pollution from region i ends up the next day at region j. A fraction qi = 1−Σj qij > 0 goes into the atmosphere and escapes. Let w (n)
i be the amount of pollution in area i after n days.
(a) Show that w (n) = u + uQ + • • • + uQn−1.
(b) Show that w (n) →w, and show how to compute w from u.
(c) The government wants to limit pollution levels to a prescribed level by prescribing w. Show how to determine the levels of pollution u which would result in a prescribed limiting value w.