Wiring up a Net-Zero Home

A homeowner decides to make their house carbon-neutral. They place solar panels on the roof, which then connect o their mains circuit via a wire.

The wire can be modelled as leaving the solar panels at A = 2i + 3j + 4k, and connecting to the mains at the point B = -3i + j – 3k, with the distances measured in metres and both points measured relative to the same fixed origin.

a) Show that \(\overrightarrow{\text{AB}} = – 5\mathbf{i -}2\mathbf{j -}7\mathbf{k}\) and hence find the length of wire needed to 2 decimal places.

[2 marks]

In many homes powered by solar energy, when excess power is generated, it can be put onto the national grid, so that more renewables power the grid rather than fossil fuels. 

The wire leading to the grid is on an automatic switch system M, which divides the wire \(\overrightarrow{\text{AB}}\) in the ratio 2:1.

b) Calculate the distance of the automatic switch system M from the origin.

[4 marks]


Mountainside Monitoring

3 CO₂ monitors K, L and M are placed on a mountain side

The vector \(\overrightarrow{\text{KL}} = 3\mathbf{i -}6\mathbf{k}\) and \(\overrightarrow{\text{LM}} = 2\mathbf{i} + 5\mathbf{j} + 4\mathbf{k}\), relative to a fixed origin.

Show that \(\angle KLM = 66.4°\) to one decimal place

[7 marks]

Hence find \(\angle LKM\) and \(\angle LMK\)

[3 marks]

A Climate Aware Citizen

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

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

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

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 CO2 emissions measured in x109 kg year-1








Rate of CO2 emissions (x109 kg year-1)







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]

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]

Storing Sequestrates

A manufacturer produces a tank for storing liquid CO2 underground.

The tank is modelled in the shape of a hollow vertical circular cylinder closed with a flat lid at the top and a hemispherical shell at the bottom. 

The walls of the tank are assumed to have negligible thickness.

The cylinder has a radius r metres and height h metres and the hemisphere has radius r metres.

The volume of the tank is 6 m3.

a) Show that, according to the model, the surface area of the tank, in m2 is given by

\[\frac{12}{r} + \frac{5}{3}\pi r^{2}\]

[4 marks]

The manufacturer needs to minimise the surface area of the tank, to minimise costs. 

b) Find, using calculus, the radius of the tank for which the surface area is a minimum.

[4 marks]

c) Find the surface area of the tank for this radius, giving your answer to the nearest integer.

[2 marks]

A Total Emissions Goal

The warming caused by carbon dioxide (CO₂) emissions over any given period is proportional to the total amount of CO₂ emitted over that period. Recognising this, a country decides to limit its carbon dioxide emissions to less than 12×10¹²kg in total emitted across 20 years.

For the first 4 years, the countries emissions are stable at 800×10⁹kg year⁻¹

The country decides that they will be able to reduce their emissions so that each subsequent year produces 5% less emissions than the previous year.

Using the model,

a) Show that the country’s total CO₂ emissions from the
first 6 years are estimated to be 4682×10⁹kg CO₂

[2 marks]

Show that the estimated total emissions per year in the rᵗʰ year, with units
x10⁹kg year⁻¹, for 5 ≤ r ≤ 20, is

800 x 0.95ʳ⁻⁴

[1 mark]

Determine whether the country will meet their 20 year emissions goal.

[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]