A Carbon Conscious Company

The curve

\[y = e^{-0.5x} + 4x – 0.1x^{2} + 2\]

Can be used to describe a company’s net emissions, in tons of CO2, x decades after 2000. 

a) By first using calculus, show that peak emissions occur at 19 ≤ x ≤ 21

[4 marks]

The company owns an afforestation program. As trees absorb CO2, this means that their net emissions can decrease to negative values.

b) Show that the emissions become negative between 39 and 41 years after the year 2000.

[2 marks]

c) Use the Newton-Raphson method to find, to 4 significant figures, the number of decades it takes for the emissions to become negative. Use x0 = 39

[4 marks]

Arithmetic Afforestation

A country decides to begin a reforestation program, starting in 2020, gradually increasing the number of trees planted per year by the same amount each year.

The timetable for the first four years is shown below:

Year: 2020 2021 2022 2023
Trees planted: 6.0×10 1.4×10 2.2×10 3.0×10

a) Find an expression, in terms of n for the number of trees planted in year n.

[2 marks]

b) Calculate how many trees in total will be planted if this program is followed for 10 years.

[2 mark] 

The government will declare the program a success once 2.45×10 trees have been planted in total

c) Given that the country plants all of the trees that the model would predict in year k, but reaches the target part way through year (k + 1), show that k satisfies 

(2k – 49)(k + 25) < 0.

[5 marks]

Profiting from Reforestation

A small company is planting trees in areas of the Amazon rainforest that have been affected by forest fires.

On any day, the cost to the company, $y, of planting x trees is modelled to be the sum of two elements:

  • a fixed cost
  • a cost that is proportional to the number of trees planted that day

a) Write down a general equation linking y with x for this model.

[2 marks]

The company is subsidised $2 by the government for each tree planted.

On a day when 80,000 trees are planted, the company makes a profit of £500

On a day when 30,000 trees are planted, the company makes a loss of £80

Using the above information,

b) Show that \(y =  0.84x + 428\)

[3 marks]

c) With reference to the model, interpret the significance of the value 0.84 in the equation.

[1 mark]

d) Find the least number of trees that must be planted on any given day for the company to make a profit that day.

[3 marks]

Rainforest Reforestation

One way of reducing carbon dioxide in the atmosphere and slowing global warming is to plant trees which absorb carbon dioxide from the atmosphere.

A scientist plants some trees in the Amazon rainforest and the Tongass rainforest.

The scientist compares the carbon dioxide absorbed from the atmosphere by the trees she planted in the Amazon rainforest with the trees she planted in the Tongass rainforest.


Carbon absorbed per tree

\(\frac{1}{5}\) more than in the Tongass

Number of trees planted

\(\frac{1}{4}\) less than in the Tongass

By what fraction is the total carbon dioxide absorbed by the trees planted in the Amazon compared with the total carbon dioxide absorbed by the trees planted in the Tongass rainforest? 
[3 marks]