Level: A Level

Wiring up a Net-Zero Home

Post date 30 September 2021

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 Hurricane Direction Shift

Post date 30 September 2021

The movement of a hurricane is modelled by vector h.

It is moving at a speed of 20 kph, with the direction of 165˚ above the equator, when portrayed on a flat map of the Earth.

a) Write h in component form.

[2 marks]

The hurricane makes landfall. Its movement is now modelled by vector l, \(\left( \frac{15\sqrt{3}}{2}\mathbf{\text{i}},\mathbf{\ }\frac{15}{2}\mathbf{j} \right)\).

b) Find the amount by which the hurricanes speed has decreased and state the hurricanes new direction.

[3 marks]

Monitoring Currents

Post date 30 September 2021

In this question, all distances are measured in kilometres.

2 deep sea ocean current monitors, A and B, have position vectors (-1, 7, k) and (4, 1, 10) respectively, relative to a fixed origin. Given that the distance from A to B is \(5\sqrt{5}\)km,

a) Find the possible values of the constant k.

[3 marks]

b) For the larger value of k, find the unit vector in the direction of \(\overrightarrow{\text{OA}}\)

[3 marks]

Mountainside Monitoring

Post date 30 September 2021

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]

Colliding Currents

Post date 30 September 2021

2 deep sea ocean currents meet.

By modelling one current as the positive y axis.

a) Find the angle that the second current, with vector \(4\mathbf{i – j}\), makes with the first current when they meet.

[3 marks]

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]

Finding the Total Carbon Dioxide Emissions from 1850

Post date 30 September 2021

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]

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