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Tree Tops

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

There were several answers to this problem but they were not solutions or explanation of a strategy so I have not included them. There are many possible solutions to this problem, which requires some sense making as well as some calculating. I therefore liked the solution from Laura of Teesside High School very much indeed. It showed some analysis and some good common sense (a very useful talent for a mathematician to have). This is followed by the solution sent in by Andrei of School 205 Bucharest.

To maximise the profit from the crop of trees both the initial expenditure and the long-term profit must be considered.

The Lodgepole Ping is the most expensive and the European Larch the cheapest of the trees. Then this is offset against the profit gained from the thinnings at both 10 and 20 years:

Sitka Spruce: -120000+10000+40000= -70000
European Larch: -158000+15000+40000= -60000
Lodgepole Pine: -130000+20000+30000= -80000

So over the first 20 years the European Larch loses the least money. However, to discover the best tree to plant for a long term profit over 70 years this loss must be taken from the profit gained when the trees are felled:

SS: 1126800-70000= 1056800
EL: 1158000-60000= 1098000
LP: 1144000-80000= 1064000

So after 70 years the European Larch makes the most profit. However after 70 years two of the trees begin to be worth less so if the trees are to be felled at 90 years the profit must be recalculated.

SS: 805000-70000=735000
EL: 837000-60000=777000
LP: 1476000-80000=1396000

So after 90 years the Lodgepole Pine is the best value, surpassing the profit gained by the European Larch after 70 years, though not by a great deal.

So perhaps for a more long term plan it would be better to plant a mixture of both Lodgepole Pine and European Larch, to reduce the initial loss and to take some revenue from the Larch after 70 years, and then replant these fields with some of the profits, then taking a larger profit at 90 years. This would give a cycle so that some money would be collected after a shorter period of time to sustain the plantation, as the extra £298000 gained with the Lodgepole Pine must be compared with the twenty years extra for this gain.

***

I observed that in the table the possible income per hectare after a number of years was written. I created a table with the profit for each type of tree, taking into account the planting cost per hectare, the profit from the thinning after 10 and 20 years (per hectare too), and the possible income after the period of time written above.

After Years
Profit (£)
Sitka Spruce
European Larch
Lodgepole Pine
30
288000
132000
42500
40
443000
409200
286400
50
623000
798000
566000
60
764000
1124000
870200
70
1056800
1098000
1064000
80
834000
999000
1230800
90
735000
777000
1396000
100
596000
726000
1280000

Now, I made a table for 70 years, with each type of tree and with possible combinations, with the number of years left and the profit obtained.

Sitka Spruce
European Larch
Lodgepole Pine
Profit (£)
30
40
0
697200
30
0
40
574400
40
30
40
418400
0
40
30
451700
70
0
0
1056800
0
70
0
1098000
0
0
70
1064000
30+40
0
0
731000
0
30+40
0
541200
0
0
30+40
328900

The manager should plant the European Larch for 70 years, obtaining a profit of £1098000 per hectare.

I made a similar table for 90 years:

Sitka Spruce
European Larch
Lodgepole Pine
Profit (£)
90
0
0
735000
0
90
0
777000
0
0
90
1396000
30+30+30
0
0
864000
0
30+30+30
0
396000
0
0
30+30+30
127500
40+50
0
0
1066000
0
40+50
0
1207000
0
0
40+50
852400
60+30
0
0
1052000
0
60+30
0
1256000
0
0
60+30
912700
30
30
30
426500
40
50
0
1241000
40
0
50
1009000
50
40
0
1032200
0
40
50
975000
0
50
40
1084400
50
0
40
909400
60
30
0
896000
60
0
30
806500
0
60
30
1166500
30
60
0
1412000
0
30
60
1002200
30
0
60
1158200

The manager should plant the Sitka Spruce for 30 years, and then the European Larch for 60 years.