A Regrowing Reef

a) Use the substitution \(u = 4 – \sqrt{s}\) to show that

\[\int_{}^{}\frac{\text{dh}}{4 – \sqrt{s}} = – 8\ln\left| 4 – \sqrt{s} \right| – 2\sqrt{s} + k\]

where k is a constant

[6 marks]

 

A coral reef is growing back after global temperatures are reduced from their peak value.

The rate of change of area covered by the reef is modelled by the differential equation

\[\frac{\text{ds}}{\text{dt}} = \frac{t^{0.25}(4 – \sqrt{s})}{20}\]

Where s is the surface area of the reef in m2 and t is the time, in years, after the reef begins to regrow.

b) Find, according to the model, the range of areas that could be covered by the coral reef. 

[2 marks]

The coral reef has a surface area of 1m2 when it starts to regrow.

According to the model,

c) Calculate the time this reef would take to cover 12 m2, giving your answer to 3 significant figures.

[7 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]

Deep-Sea Bubbles

Deep sea vents can emit harmful gases, such as hydrogen sulphide. Since these bubbles are small, they shrink once they leave the vent, as the gases dissolve into the ocean.

An autonomous submarine takes a photograph of these bubbles that looks like this graph sketch. 

deep-sea bubbles

A scientist discovers that the surface of the bubbles visible in this photograph can be represented by the curve:

\[\sin\frac{y^{3}}{100} + e^{- x}y + x^{2} = 5\ ,\ y > 5\]

By first using implicit differentiation to show that

\[\frac{\text{dy}}{\text{dx}} = \frac{100(y – 2e^{x}x)}{3e^{x}y^{2}\cos\frac{y^{3}}{100} + 100}\]

Show that at the turning points on the curve, where \(\frac{\text{dy}}{\text{dx}} = 0\), \(y = 2xe^{x}\)

[8 marks]

Filling a Weather Balloon

The height (h km) that a weather balloon can reach is related to the volume (v m₃) of helium in it at sea level by the equation:

\[h = \frac{8v^{2}}{5} – \frac{32v^{3}}{255}\ ,\ v \leq 12\]

a) Find the volume of helium required to achieve the maximum height and state this height.

[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]

Internal combustion engines

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]

Shrinking Species

Since 1800, the number of amphibian species, N, has been decreasing over time, t. 

A simple model shows that the rate of decrease of the number of species is proportional to the remaining number of species.

Given that the initial number of amphibian species is N0, and t is the number of years since 1800,

a) Show that \(N = N_{0}e^{- kt}\)

[4 marks]

In 2000 the number of amphibian species is 0.9N₀.

b) Find the exact value of k.

[3 marks]

c) Using the model, in what year will 20% of amphibian species be
extinct?

[3 marks]

The Swelling Sahara

Human-induced global warming is causing deserts such as the Sahara to increase in surface area.

In 1950 the area of the Sahara Desert was 9,200,000 km2, whereas in 2000 the area of the Sahara Desert had increased to 9,930,000 km2 due to human-induced global warming.

A model could be used to relate the surface area of the Sahara Desert, S km2, to the time, t, in years since 1950.

a) By first forming an exponential model for the surface area of the Sahara Desert relating S and t, show that the increase in the surface area of the Sahara Desert is approximately 0.15% per year.

[5 marks]

b) Use the model formed in (a) to estimate the size of the Sahara Desert in 2050. Give your answer in km2 to 3 s.f.

[1 mark]

A Shrinking Rainforest

  1. In a simple model, the surface area, S km2, of a shrinking rainforest depends on the time, t, in years since 1980.

The following information is available for rainforest A.

  • its surface area in 1980 was 300,000 km2
  • its surface area in 1981 was 294,000 km2
a) Use an exponential model to form, for rainforest A, a possible equation linking S with t.

[4 marks]

The surface area of rainforest A is monitored over a 30-year period. Its surface area after 30 years is 150,000 km2.

b) Evaluate the reliability of your model in light of this information.

[2 marks]

The following information is known about rainforest B.

  • it had the same surface area, in 1980, as rainforest A
  • it is harder to access by road, so the rate of deforestation is less than rainforest B and its surface area decreases more slowly than that of rainforest A
c) Explain how you would adapt the equation found in (a) so that it could be used to model the surface area of rainforest B.

[1 mark]

A Speedy Sea Current

An ocean current separates into 2 different currents at a small island that can be modelled as the origin. Current A heads due south and current B heads on a bearing of 100˚. 

An oceanographer wants to measure the relative surface speeds of these two separate currents, by placing buoys in them off the island and measuring the distance travelled by the buoy in 20 hours.

After 20 hours, the buoy in current A has travelled 150km whilst the buoy in current B has travelled 250km.


a)
 Find the average speeds of the two ocean currents.

[2 marks]

b) Calculate the final distance between the buoys.

[2 marks]

c) Θ is the final bearing of the buoy in current B from the buoy in current A. Show that \(\frac{\sin\theta}{\sin{80}} = 0.93\) to 2 decimal places.

[3 marks]

d) Find the area of the triangle with the island and the two buoys as its
vertices.

[3 marks]