Tree tops
Problem
Imagine you have a large plot of land ideal for planting a forest.
First you plant the trees...
Click below to find out the costs of planting.
Terry's Trees
For all your planting needs
We bring the saplings and do all the planting so you don't have to!
Sitka Spruce £120 000 per hectare
European Larch £115 000 per hectare
Lodgepole Pine £130 000 per hectare
Give us a call and we'll do the rest!
Then, after 10 years, the forest is thinned to give the remaining trees more room to grow.
The forest needs to be thinned again after another 10 years.
The wood from the thinning is sold. Click below to see the rates.
Theo's Thinning Company
Got a forest that needs thinning? Call Theo!
We pay the best rates:
| 10 year thin* | 20 year thin* | |
| Sitka Spruce | 10 | 40 |
| European Larch | 15 | 40 |
| Lodgepole Pine | 20 | 30 |
*All values quoted are thousands of £ per hectare
Finally, the time will come when you cut down all the trees in the forest and sell the timber. You might decide to do this 30 or 40 years or even longer after first planting your trees! Nobody said this was a "get rich quick" scheme...
Click below to see how much profit you will make from selling your timber, depending on when you choose to cut down your trees.
Lou's Lumber
Got a forest you no longer need?
We can cut it down!
Whether your trees are 30 years old or 100 years old, we will pay the best prices.
Check the table below for our rates.
| Age of Forest (years) |
Sitka Spruce |
European Larch |
Lodgepole Pine |
| 30 | 358 | 192 | 123 |
| 40 | 513 | 469 | 366 |
| 50 | 693 | 858 | 646 |
| 60 | 834 | 1184 | 950 |
| 70 | 1127 | 1158 | 1144 |
| 80 | 904 | 1059 | 1311 |
| 90 | 805 | 837 | 1476 |
| 100 | 666 | 783 | 1360 |
All prices are quoted in thousands of £ per hectare.
What strategy will maximise the profits for you and your family in the decades to come?
Here are some questions you could consider
- Which tree should you choose to minimise the cost of planting the forest?
- Which tree should you choose to minimise your loss after 20 years?
- What would the profit be for each type of tree after 30, 40, and 50 years?
- What happens to the prices paid for each type of wood over the 100 year period? Why might this be the case?
What's the maximum profit you could make after 100 years?
Getting Started
This would cost £120 000.
After 10 years, you would get £10 000 from the 10-year thinning:
-120 000 + 10 000 = -110 000 so you have a loss of £110 000 after 10 years.
After another 10 years, you get £40 000 from the 20-year thinning:
-110 000 + 40 000 = -70 000 so your loss after 20 years is £70 000.
You could then choose to chop down your forest and sell the wood after another 10 years. This would bring in £358 000.
-70 000 + 358 000 = 288 000. So after 30 years, you have a profit of £288 000.
However, you could choose instead to keep the trees and cut down the forest after 40 years. This would bring in £513 000.
-70 000 + 513 000 = 443 000. So after 40 years, your profit would be £443 000.
Perhaps you could set up a spreadsheet to calculate the costs and profits for each type of tree!
If you want to maximise profits after 100 years, you could investigate planting a forest for 50 years, and then cutting it down and planting another 50-year forest.
Or planting a 60-year forest and a 40-year forest.
Or two 30-year forests and a 40-year forest.
Or...
Student Solutions
- Which tree should you choose to minimise the cost of planting the forest?
In the first place European Larch is the cheapest tree to build when you start your farming because it cost the lowest.
- Which tree should you choose to minimise your loss after 20 years?
After 10 years and 20 years, we need to thin the trees.
Rates:
SS: +50k
EL: +55k
LP: +50k
The total cost for planting and thinning is:
SS: -70k
EL: -60k
LP: -80k
- What would the profit be for each type of tree after 30, 40, and 50 years?
After 30 years, the Sitka Spruce will make the most profit
After 40 years, the Sitka Spruce will make the most profit
After 50 years, the European Larch will make the most profit
After 60 and 70 years, the European Larch will make the most profit.
After 80, 90, 100 years, the European Larch and the Sitka Spruce lost value, the Lodgepole Pine will make the most profit.
Best in 0,10,20,50,60,70: European Larch
Best in 30,40: Sitka Spruce
Best in over 80: Lodgepole Pine
- What's the maximum profit you could make after 100 years?
I saw that if you keep a pine tree for 90 years you get £1,476,000 which was the largest amount. Then you'd add [ £50,000] (money gained from thinning at 10 and 20 years) then you'd take away £130,000 (cost of planting) giving you [ £1,396,000] per hectare.
For the [other trees], using the above method you would select the highest amount then and [add the thinning price], then take away the [planting] price.
Ziniu found the profit per year to find a better strategy:
To find the optimal rates, we need to divide the amount of money gained by the years it takes to grow the trees to that stage so we can find the most efficient time/tree type to maximise profits. We need to also factor in the cost of planting and the gains of thinning as that may affect our results.
Calculation:
(amount gained from cutting + money gained from thinning - planting costs) divided by years it took to plant to that stage
Amount shown is in thousands of £
Tree name, after 30 years, after 40 years, etc
SS: 9.6, 11.1, 12.46, 12.73, 15.1, (no point calculating past this point as the value drops while years increase)
EL: 4.4, 10.23, 15.96, 18.73, 15.68, (no point calculating past this point as the value drops while years increase)
LP: 1.43, 7,15, 11,32, 14.5, 15.2, 15.39, 15.51 (no point calculating past this point as the value drops while years increase)
The most time and money efficient planting method is European Larch 60 years.
The question says to calculate the maximum amount of money earned in 100 years. 50, 20 and 10 are all multiples of 100. Planting and cutting trees at that time would maximise the efficiency of the plantation.
The most efficient plantation method would be EL 50 years, gaining 15960 £
a year and 1596000 £ after 100 years.
As well as replanting after chopping down the forest, Edward came up with a clever and detailed strategy involving planting different kinds of trees at the same time. Click here to watch Edward's video. Note that Edward has interpreted the thinning process in a different way.
Teachers' Resources
Why do this problem?
This problem offers opportunities for students to interpret data in a real life context and process information in order to answer questions. There is quite a lot of freedom in how to approach the task, so students will need to make decisions on how to organise their working.
As part of the "Evaluating Methods, Improving Solutions" feature, this problem encourages students to find a value for the profit after a certain number of years, and then look for ways of improving on it by working within the constraints of the situation. We hope this challenge will engage students' curiosity and encourage them to persevere until they find the optimum
solution.
Possible approach
Set the scene for the problem:
"Imagine you have a large plot of land ideal for planting a forest. First you plant the trees. Then, after 10 years, the forest is thinned to give the remaining trees more room to grow. The forest needs to be thinned again after another 10 years. The wood from the thinning is sold. Finally, the time will come when you cut down all the trees in the forest and sell the timber. You
might decide to do this 30 or 40 years or even longer after first planting your trees! In this lesson, we're going to try to find the best strategy for maximising profits for your forestry company."
Hand out the resource sheet (perhaps cut into three separate sections - Terry's Trees, Theo's Thinning Company, and Lou's Lumber) and invite students to think about the following:
"What sort of questions occur to you that it might be useful to answer in order to find a good strategy?" Give them some time to discuss with their partner, and then bring the class together and write up the questions on the board. Here are some examples of the sort of questions that might emerge:
Which tree is the cheapest to plant?
Which tree minimises the loss after twenty years?
What would the profit be for each type of tree after 30, 40, and 50 years?
What happens to the prices paid for each type of wood over the 100 year period? Why might this be the case?
What's the maximum profit you could make after 100 years?
Then set the class to work on some of their questions. Some students may wish to draw tables by hand and use calculators. Others might prefer to use a computer and create a spreadsheet with the data, and use formulae to calculate the profit for each type of tree and each timescale. As the class are working, circulate to see the different approaches people are using and the different strategies
they come up with for organising the data. It might be appropriate to bring the class together to share some of these strategies, so that students can learn from each other's approaches.
At the end of the lesson, set aside some time for students to explain the strategy they would recommend for planting the forest, together with their reasoning.
Key questions
Does it help to put the information in a table?
Are there any diagrams or graphs that could be used?
Is a spreadsheet useful?
Possible support
The Getting Started section goes through the calculations for the first forty years for the Sitka Spruce. Students could use this example to repeat the calculations with the other two tree types, and make comparisons.
Possible extension
Students might begin by exploring each type of tree separately. Encourage them to consider the possibility of combining trees - for example, if they try to maximise profits over 100 years, they could plant one type of tree for 60 years and then a different type for the next 40 years. Students could then work towards a strategy for managing a forest with a cycle of planting, thinning and harvesting, that would give a sustainable and regular profit over time.