Mathematical Topic: Numerical Methods

A Climate Aware Citizen

Post date 30 September 2021

A person decides in 2020 that they want to completely eradicate their carbon footprint in 20 months.

Following this decision, they begin to use multiple technologies that decrease their carbon footprint, such as limiting air travel, carbon offsetting and a solar powered home that contributes energy back to the national grid.

This results in their rate of emissions per month, E, following the equation

\[E = 12\ln\left( t + 5 \right) – 2t + 2\]

where t is the time, in months, since 2020.

a) Show that they achieve zero emissions between 18 and 22 months after they start.

[2 marks]

b) Using the iteration formula \(t_{n + 1} = 6\ln{(t + 5)} + 1\) with \(t_{o} = 18\), find the value of \(t\) at which they achieve zero emissions to 2 decimal places.

[3 marks]

City Emission Levels

Post date 30 September 2021

The emissions of a city from 2000 to 2012 are modelled by the equation

\(p\left( t \right) = \frac{1}{10}\ln\left( t + 1 \right) – \cos\frac{t}{2} + \frac{1}{10}t^{\frac{3}{2}} + 199.3\)

\[0 \leq t \leq 12\]

a) Show that the emissions reach a local maximum in the interval \(8.5 \leq t \leq 8.6\)

[5 marks]

The emissions reach a local minimum between 9 and 11 years after the measurements began.

b) Using the Newton-Raphson procedure once and taking \(t_{0} = 9.9\) as a first approximation, find a second approximation of when the emissions reach a local minimum.

[6 marks]

A Carbon Conscious Company

Post date 30 September 2021

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]

UK Carbon Dioxide Emissions

Post date 30 September 2021

The rate of CO₂ emissions for the UK was measured every 5 years, from 1990 to 2015.

The results are given in the table below with the rate of CO₂emissions measured in x10⁹kg year^-1

Year	1990	1995	2000	2005	2010	2015
Rate of CO₂emissions (x10⁹ kg year^-1)	595.7	557.5	555.7	555.2	496.7	403.8

Using all of this information,

a) Estimate the total CO₂ emissions from the UK between 1990 and 2015, giving your answer in standard form.

[3 marks]

Given that the curve produced by plotting a graph of the rate of CO₂ emissions against the year is concave,

b) Explain whether your answer to part (a) is an underestimate or an overestimate of the total CO₂ emissions between 1990 and 2015.

[1 mark]

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