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Level: A Level

A Solar Sine Curve

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 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

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]

Arithmetic Afforestation

Arithmetic Afforestation

A country decides to begin a reforestation program, starting in 2020, gradually increasing the number of trees planted per year by the same amount each year.

The timetable for the first four years is shown below:

Year: 2020 2021 2022 2023
Trees planted: 6.0×10⁶ 1.4×10⁷ 2.2×10⁷ 3.0×10⁷

a) Find an expression, in terms of n for the number of trees planted in year n.

[2 marks]

b) Calculate how many trees in total will be planted if this program is followed for 10 years.

[2 mark] 

The government will declare the program a success once 2.45×10⁹ trees have been planted in total

c) Given that the country plants all of the trees that the model would predict in year k, but reaches the target part way through year (k + 1), show that k satisfies 

(2k – 49)(k + 25) < 0.

[5 marks]

EV Increase

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]

A Total Emissions Goal

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 Nuclear Disaster

A Nuclear Disaster

Chernobyl is the site of a nuclear disaster that happened in 1986.

Due to the slow decay of radioactive elements, the site is still extremely dangerous today.

In 2016, engineers covered the main damaged nuclear reactor with a giant concrete cap, as shown below:

A NUCLEAR DISASTER

The cross section of this cover can be modelled with the parametric equations:

 

\[x = 8\left( t + 10 \right)\ ,\ y = 100 – t^{2}\ ,\ – 19 \leq t \leq 10\]

 

a) Find the Cartesian equation of this model.

[3 marks]

b) The width of the cap, according to the model.

[2 marks]

An Advancing Hurricane

An Advancing Hurricane

A hurricane is heading from the Atlantic Ocean to Miami, Florida. If the centre of the hurricane goes within a 50 kilometre radius of the city centre, the city will sustain very severe damage.

A meteorologist models the city centre as a fixed particle located at the point (240 , 200). They then model the centre of the hurricane as travelling on a path relative to the city defined by \(y = kx + 20\). 

The unit of distance for the meteorologist’s coordinate system is kilometres.

 

a) Find the values of k such that the city does not sustain severe damage.

[8 marks]

b) State one limitation of the model.

[1 mark]

A Carbon Negative Company

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]

Separated Satellites

Separated Satellites

2 weather-monitoring satellites orbit the earth.

One is in a circular orbit C₁, the other orbits in an extreme ellipse, C₂ so that it can get closer to the surface. 

These orbits can be modelled by the equations below:

\[C_{1}:\left( x + 2\sqrt{17} \right)^{2} + y^{2} = 66\]

\[C_{2}:x = 10\cos t,\ y = 4\sqrt{2}\sin t,\ 0 \leq t \leq 2\pi\]

Give the x coordinate of the points of intersection of the curves C₁ and C₂, given that

\(- 5 \leq x \leq 0\). Give an exact answer in the form

\[A\sqrt{1122} + B\sqrt{17}\]

[7 marks]

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