Monitoring Currents

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

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

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 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]

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 Ferocious Forest Fire

During a particularly hot summer, the area of a small forest was reduced by fire. The area, Akm2, of the surviving forest is modelled as being inversely proportional to t, the time in days since the start of the fire.

a) 5.5 days after the fire started, the area of the surviving forest was 2.6 km2. Find the constant of proportionality, k.

[2 marks]

b) Sketch the graph of A against t, stating the equations of any asymptotes.

[2 marks]

c) Suggest one reason why this model has the restriction t ≥ 1

[1 mark]

d) Suggest a refinement to this model that might make it more accurate

[1 mark]

Profiting from Reforestation

A small company is planting trees in areas of the Amazon rainforest that have been affected by forest fires.

On any day, the cost to the company, $y, of planting x trees is modelled to be the sum of two elements:

  • a fixed cost
  • a cost that is proportional to the number of trees planted that day

a) Write down a general equation linking y with x for this model.

[2 marks]

The company is subsidised $2 by the government for each tree planted.

On a day when 80,000 trees are planted, the company makes a profit of £500

On a day when 30,000 trees are planted, the company makes a loss of £80

Using the above information,

b) Show that \(y =  0.84x + 428\)

[3 marks]

c) With reference to the model, interpret the significance of the value 0.84 in the equation.

[1 mark]

d) Find the least number of trees that must be planted on any given day for the company to make a profit that day.

[3 marks]

Surviving Species

Climate change affects the habitats and environments of many species, some of which won’t be able to adapt fast enough to survive in their new habitats.

The graph shows the percentage of species driven extinct since 1500. Of the species that were around in 1500

Diagram showing Extinctions since 1500

a) Calculate the probability of a reptile species having gone extinct by 1900. 

[1 marks]

b) Calculate the probability of an amphibian species not having gone extinct by 2018. 

[1 marks]

c) Of a sample of 60,000 species alive in 1500, assuming equal numbers of amphibian, mammal, bird, reptile and fish species are included, find, by first taking an average, how many species you would expect to have not gone extinct by 2018.

[3 marks] 

Reducing Biodiversity Loss

The graph from the IPBES Global Assessment Report on Biodiversity and Ecosystem Services shows 3 different scenarios for how we could reduce biodiversity loss by 2050. Each scenario prevents the same amount of biodiversity loss.

graph from the IPBES Global Assessment Report on Biodiversity and Ecosystem Services

a) Look at the Global Technology scenario. What is the biggest measure that would be taken in this scenario? 

[1 mark]

b) Look at all three scenarios together. Which scenario would involve reducing infrastructure expansion the most? 

[1 mark]

c) What measure would be taken in the Consumption Change scenario that would not be taken in the scenarios of Global Technology or Decentralised Solutions? 

[1 mark]

d) Using a ruler, work out the percentage decrease of increasing agricultural productivity when comparing the scenario of Global Technology to Decentralised Solutions. 

[3 marks]

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