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]

A Carbon Negative Company

A small company’s carbon dioxide (CO₂) emissions since 2000 can be modelled using the parametric equations

\[12x = t,\ y = 8t – 4.9t^{2} + 10,\ t \geq 0\]

Where x is the number of years since 2000 and y the yearly CO₂ emissions, in tonnes. 

Due to an afforestation program, the company’s emissions can go negative, as the trees planted absorb CO₂ from the atmosphere. 

a) Find the year in which the company’s CO₂ emissions go negative.

[4 marks]

b) Find the greatest amount of CO₂ emissions from that company in one year

[5 marks]