Finding the Total Carbon Dioxide Emissions from 1850

The graph below shows how the rate of CO2 emissions varies from 1800 to 2017.

This curve can be approximated as \(E = 1.5e^{0.02t}\) where E is the rate of CO₂ emissions per year (GtCO₂ year⁻¹) and t is the number of years since 1800.

a) Using the above equation, calculate the total mass of CO₂ emitted between the years 1800 and 2017. Give your answer to 3.s.f.

[3 marks]

b) By comparing the exponential model with the best fit line on the graph shown at \(t = 150\), evaluate whether your answer to (a) is an overestimate of the true emissions or an underestimate.

[1 mark]

A Regrowing Reef

a) Use the substitution \(u = 4 – \sqrt{s}\) to show that

\[\int_{}^{}\frac{\text{dh}}{4 – \sqrt{s}} = – 8\ln\left| 4 – \sqrt{s} \right| – 2\sqrt{s} + k\]

where k is a constant

[6 marks]


A coral reef is growing back after global temperatures are reduced from their peak value.

The rate of change of area covered by the reef is modelled by the differential equation

\[\frac{\text{ds}}{\text{dt}} = \frac{t^{0.25}(4 – \sqrt{s})}{20}\]

Where s is the surface area of the reef in m2 and t is the time, in years, after the reef begins to regrow.

b) Find, according to the model, the range of areas that could be covered by the coral reef. 

[2 marks]

The coral reef has a surface area of 1m2 when it starts to regrow.

According to the model,

c) Calculate the time this reef would take to cover 12 m2, giving your answer to 3 significant figures.

[7 marks]

50 Years to Net-Zero

A country wishes to achieve net-zero CO2 emissions in 50 years. 

At the start of the program their emissions are 800MtCO2 year-1. 

They decide that they will be able to reduce their emissions at a stable rate so that each subsequent year they emit 12MtCO2 less than the previous year.

a) Calculate the total emissions that the country had produced over the 50 years, giving your answer in MtCO2.

[2 marks]

b) Show that a graph of MtCO2 produced per year against the year follows a straight line with equation:

\[y = 800 – 12x\]

[1 mark]

At the same time as reducing their emissions, the country decides to start a carbon dioxide removal program, whereby a certain amount of carbon dioxide is captured from the atmosphere and sequestered underground each year. 

The program begins in the tenth year. 

When the graph of MtCO2 removed per year is plotted against the year, it follows the curve with equation

\[y = 0.1x^{2} – x\]

c) Determine whether the country achieves their goal by finding the year in which the emissions removed are equal to the emissions produced, and thus the net emissions from the country are zero.

[3 marks]

After the 50 year program, the countries emissions stabilise at the final value. 

The MtCO2 absorbed per year follows the same trend as before. 

The country wishes to have not contributed to global warming at all since the start of the program. To achieve this, their net total CO2 emissions over the entire program would have to be zero.

d) Given the above information, by using calculus show that it takes 109 years for the country to have had a net zero effect on global warming since the start of the study.

[5 marks]