Pollution Problem

Recurrence Relations

A factory wishes to dispose of waste into a sea loch. They wish to dump 4 tonnes of waste into the loch at the end of each week.

It is estimated that the flushing action of the tide will wash away 25% of this waste each week.

The initial amount of waste in the loch is 2 tonnes.It is estimated that the maximum tonnage of waste,

which can be supported by the loch ecosystem, is 18 tonnes. It is your job to decide if the company's proposals are

safe or what modifications are required to their proposals for them to be safe.

A Pollution Problem

Setting up the recurrence relation:

   waste at waste at waste waste   start of = start of - flushed + added by next week this week out factory

i.e. un = un-1 - 25% un-1 + 4

which simplifies to give un = 0.75un-1 + 4 with u0 = 2.

1. On the [MODE] screen select Seq mode and Dot mode.

2. Press [Y=] and using the cursor keys. Set nMin to zero. Enter the recurrence relation in u(n) =u is its own key, [u] which is above [7]n is [ X,T,q,n ] when in Seq mode .

Set u(nMin) = 2. This is the initial amount of waste in the loch.

3. Press [WINDOW]Set nMin to zero.Still using the cursor keys move to nMax . Enter a value of 60 to define 60 weeks of this cycle .Set PlotStart and PlotStep to 1. Set the other WINDOW variables as follows Xmin = 0 Xmax = 60 Xscl = 2 Ymin = 0 Ymax = 24 Yscl = 2

4. Press [GRAPH] .The graph of the recurrence relation is drawn as a sequence diagram .The graph can be examined using the [TRACE] button and the cursor keys

5. Use the table set-up, [TBLSET] to set the initial values so that TblStart = 0 and Δ Tbl = 1. Use [TABLE] to produce the table of values generated by the recurrence relation.

The sequence tends to a limit of 16. From this result you would deduce that the loch's ecosystem would be 'safe' if the proposals stated for dumping went ahead.

6. Try changing the value of u(nMin) which would simulate different estimates of the initial amount of waste in the loch. It is also easy to see how the pollution levels can be affected by alterations to other factors e.g. flushing action being better or worse than estimated, by changing the recurrence relation. You can enter up to three recurrence relations on the TI-83 at any one time. This can be done by also using the [v] and [w] buttons, which are located above the [8] and [9] respectively.You may wish to use this facility if you want to compare the sequences produced by two different recurrence relations. For example, suppose the chemical manufacturer in our example made a late request to allow him to dump an extra tonne per week.

The recurrence relation describing this situation would be vn = 0.75vn-1 + 5 and this is entered as shown in the diagram. The sequence diagrams and the tables are obtained as before and are shown in the diagrams below.

By examining the table of values produced and the sequence diagram you should find that this proposal would not be allowed to go ahead (it would damage the ecosystem of the loch).