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]


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]

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]

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]

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]

Ice Incineration

A country’s government wants to reduce the number of cars using internal combustion engines by encouraging the purchase of electric vehicles.

The total number of cars using internal combustion engines in a country at time t can be modelled by the equation

\[N\left( t \right) = \left( 3.5 \times 10^{7} + E \right)e^{- \frac{1}{10}t} – E\]

Where N(t) is the number of cars using internal combustion engines at time t years after the program was introduced and E is the number of electric vehicles on the roads before the government initiative.

Using the equation for this model,

a) Explain the significance of the number 3.5 x 107.

[1 mark]


b) Explain why the initial number of cars using internal combustion engines is independent of the initial number of electric vehicles.

[2 marks]

c) Calculate the number of cars using internal combustion engines after 10 years if the initial number of electric vehicles is 2 x 106.

[2 marks]

d) Analyse the suitability of this model by evaluating it for large values of t.

[2 marks]

A Solar Sine Curve

You are given the equation

\[f\left( x \right) = 5\cos\theta – 8\sin\theta\]

a) Express f(x) in the form \(R\cos{(\theta + \alpha})\) where \(R > 0\) and\(\ 0 < \alpha < \pi\). Write R in surd form and give the value of α correct to 4 decimal places.

[4 marks]

The temperature of a solar panel, T ˚C, can be modelled by the equation

\[T = 20 + 5\cos\frac{4x}{15} – 8\sin\frac{4x}{15}\ ,\ 0 \leq x \leq 72\]

Where x is the time in hours since 10pm one evening?

b) Calculate the maximum value of T predicted by this model and the value of x, to 2 decimal places, when this value first occurs.

[4 marks]

c) Calculate the times during the first 24 hours when the temperature is predicted, by this model, to be exactly 17 ˚C

[4 marks]

A New Renewable

A scientist wishes to develop a new way of generating renewable energy.

They decide to use a large magnet on a large spring, oscillating through a coil of wire, generating a current that could then charge a battery, which in turn would store the energy released from the system.

The system is set into motion with a winch system that can be used to set the spring. The following equation can then be used to describe the oscillations, in metres, t minutes after it has been set into motion:

\[A = 10e^{- 0.05t}\cos{10t}\]

a) On a plot of A against t, show that the t values of the
turning points satisfy the equation \(\tan{10t} = – \frac{1}{200}\)

[4 marks]

The battery can only be charged when the amplitude of the oscillations of the spring are greater than 1m. The amplitudes, A, of the oscillations are found at the turning points of the curve.

b) Give the t value of the last turning point where |A|≥1m, and the battery can no longer be charged.

[4 marks]

Net-Zero Buses

15 cities, each of varying sizes, decide to have carbon-neutral public transport systems.

When the cities are arranged in size order from smallest to largest, the number of new electric buses they need follows an arithmetic sequence with first term (representing the smallest amount of buses needed for the smallest city) a buses and common difference d buses. 

The 7ᵗʰ city in this progression needs 580 new electric buses and the largest city needs 1020 new electric buses.

Find the value of a and the value of d.

[5 marks]

EV Increase

A country decides to subsidise the purchase of electric vehicles, causing more people to buy them.

Initially, the country used an equivalent of 56 million tonnes of oil for transport.

Since the government subsidy, the transport sector in each subsequent year has required 80% of the amount of oil that the previous year required.


a) Write a recurrence relation to model the amount of oil used, in million tonnes, in each subsequent year.

[2 marks]

b) Find the amount of oil needed for the transport sector after the fifth year.

[2 marks]

c) Find the total amount of oil used until the transport sector requires no more oil.

[4 marks]

d) State one limitation with the model.

[1 mark]