Recurrence Relations

1
Recurrence Relations

• Pollution Problems

2

•  Q1. The initial quantity of pollution in the
  loch is 45 tons, the Council remove 36 during
  the week and a factory discharges 9 tons into the
  loch each Sunday.
•    i) Find the amount of pollution after 1, 2, 3
  and 4 weeks.
•  ii) Establish a recurrence relation and hence
  find the long term state of the loch.

3

• Repeat question 1 for examples 2-5 which follow.
•  Q2. Initial quantity of pollution 33 tons
•    Council remove 35 per week during the week.
•    Factory is allowed to discharge 20 tons per
  week at the end of the week. . 
•  Q3. Initial quantity of pollution 40 tons
•    Council remove 60 per week during the week
•    Factory is allowed to discharge 15 tons per
  week at the end of the week.

4

•    Q4. Initial quantity of pollution 16.5 tons
  
•    Council remove 52 per week during the week
•  Factory is allowed to discharge 6 tons per week
  at the end of the week.
•    Q5. Initial quantity of pollution 2000 tons
  
•   Council remove 34 per week during the week
•  Factory is allowed to discharge 45 tons per
  week at the end of the week.

5

• Q6. Initial quantity of pollution 380 tons
•    Council remove 15 per week during the week
  for each of the first five weeks and then they
  must remove 20 per week after this period.
•  Factory is allowed to discharge 20 tons per week
  at the end of the week.
•    Q7. Initial quantity of pollution 100 tons
•    Council remove 35 per week during the week
  for each of the first four weeks and then they
  must remove 22 per week after this period.
•  Factory is allowed to discharge 35 tons per week
  at the end of the week.

6

• Q8. Initial quantity of pollution 600 tons
•  Council remove 45 per week during the week for
  each of the first three weeks 35 per week
  during the week for each of the next four weeks
  and then they must remove 25 per week, every
  week, after this period.
•  Factory is allowed to discharge 40 tons per week
  at the end of the week.