1.06i Exponential growth/decay: in modelling context

The mass, \(m\) grams, of a substance at time \(t\) years is given by the formula $$m = 180e^{-0.017t}.$$

1. Find the value of \(t\) for which the mass is 25 grams. [3]
2. Find the rate at which the mass is decreasing when \(t = 55\). [3]

A substance is decaying in such a way that its mass, \(m\) kg, at a time \(t\) years from now is given by the formula $$m = 240e^{-0.04t}.$$

1. Find the time taken for the substance to halve its mass. [3]
2. Find the value of \(t\) for which the mass is decreasing at a rate of 2.1 kg per year. [4]

The mass, \(m\) grams, of a substance is increasing exponentially so that the mass at time \(t\) hours is given by $$m = 250e^{0.02t}.$$

1. Find the time taken for the mass to increase to twice its initial value, and deduce the time taken for the mass to increase to 8 times its initial value. [3]
2. Find the rate at which the mass is increasing at the instant when the mass is 400 grams. [3]

Oil is leaking into the sea from a pipeline, creating a circular oil slick. The radius \(r\) metres of the oil slick \(t\) hours after the start of the leak is modelled by the equation $$r = 20(1 - e^{-0.2t}).$$

1. Find the radius of the slick when \(t = 2\), and the rate at which the radius is increasing at this time. [4]
2. Find the rate at which the area of the slick is increasing when \(t = 2\). [4]

The temperature \(\theta\) °C of water in a container after \(t\) minutes is modelled by the equation $$\theta = a - be^{-kt},$$ where \(a\), \(b\) and \(k\) are positive constants. The initial and long-term temperatures of the water are 15°C and 100°C respectively. After 1 minute, the temperature is 30°C.

1. Find \(a\), \(b\) and \(k\). [6]
2. Find how long it takes for the temperature to reach 80°C. [2]

The height \(h\) metres of a tree after \(t\) years is modelled by the equation $$h = a - be^{-kt},$$ where \(a\), \(b\) and \(k\) are positive constants.

1. Given that the long-term height of the tree is 10.5 metres, and the initial height is 0.5 metres, find the values of \(a\) and \(b\). [3]
2. Given also that the tree grows to a height of 6 metres in 8 years, find the value of \(k\), giving your answer correct to 2 decimal places. [3]

The value \(£V\) of a car \(t\) years after it is new is modelled by the equation \(V = Ae^{-kt}\), where \(A\) and \(k\) are positive constants which depend on the make and model of the car.

1. Brian buys a new sports car. Its value is modelled by the equation $$V = 20000 e^{-0.2t}.$$ Calculate how much value, to the nearest £100, this car has lost after 1 year. [2]
2. At the same time as Brian buys his car, Kate buys a new hatchback for £15000. Her car loses £2000 of its value in the first year. Show that, for Kate's car, \(k = 0.143\) correct to 3 significant figures. [3]
3. Find how long it is before Brian's and Kate's cars have the same value. [3]

The mass of radioactive atoms in a substance can be modelled by the equation $$m = m_0 k^t$$ where \(m_0\) grams is the initial mass, \(m\) grams is the mass after \(t\) days and \(k\) is a constant. The value of \(k\) differs from one substance to another.

1. 1. A sample of radioactive iodine reduced in mass from 24 grams to 12 grams in 8 days. Show that the value of the constant \(k\) for this substance is 0.917004, correct to six decimal places. [1 mark]
   2. A similar sample of radioactive iodine reduced in mass to 1 gram after 60 days. Calculate the initial mass of this sample, giving your answer to the nearest gram. [2 marks]
2. The half-life of a radioactive substance is the time it takes for a mass of \(m_0\) to reduce to a mass of \(\frac{1}{2}m_0\). A sample of radioactive vanadium reduced in mass from exactly 10 grams to 8.106 grams in 100 days. Find the half-life of radioactive vanadium, giving your answer to the nearest day. [4 marks]

An entomologist is studying the population of insects in a colony. Initially there are 300 insects in the colony and in a model, the entomologist assumes that the population, \(P\), at time \(t\) weeks satisfies the differential equation $$\frac{dP}{dt} = kP,$$ where \(k\) is a constant.

1. Find an expression for \(P\) in terms of \(k\) and \(t\). [5]

Given that after one week there are 360 insects in the colony,

1. find the value of \(k\) to 3 significant figures. [2]

Given also that after two and three weeks there are 440 and 600 insects respectively,

1. comment on suitability of the modelling assumption. [2]

An alternative model assumes that $$\frac{dP}{dt} = P(0.4 - 0.25 \cos 0.5t).$$

1. Using the initial data, \(P = 300\) when \(t = 0\), solve this differential equation. [3]
2. Compare the suitability of the two models. [2]

The growth of a tree is modelled by the differential equation $$10\frac{dh}{dt} = 20 - h,$$ where \(h\) is its height in metres and the time \(t\) is in years. It is assumed that the tree is grown from seed, so that \(h = 0\) when \(t = 0\).

1. Write down the value of \(h\) for which \(\frac{dh}{dt} = 0\), and interpret this in terms of the growth of the tree. [1]
2. Verify that \(h = 20(1 - e^{-0.1t})\) satisfies this differential equation and its initial condition. [5]

The alternative differential equation $$200\frac{dh}{dt} = 400 - h^2$$ is proposed to model the growth of the tree. As before, \(h = 0\) when \(t = 0\).

1. Using partial fractions, show by integration that the solution to the alternative differential equation is $$h = \frac{20(1 - e^{-0.2t})}{1 + e^{-0.2t}}.$$ [9]
2. What does this solution indicate about the long-term height of the tree? [1]
3. After a year, the tree has grown to a height of 2 m. Which model fits this information better? [3]

Scientists can estimate the time elapsed since an animal died by measuring its body temperature.

1. Assuming the temperature goes down at a constant rate of 1.5 degrees Fahrenheit per hour, estimate how long it will take for the temperature to drop
   1. from 98°F to 89°F,
   2. from 98°F to 80°F. [2]

In practice, rate of temperature loss is not likely to be constant. A better model is provided by Newton's law of cooling, which states that the temperature \(\theta\) in degrees Fahrenheit \(t\) hours after death is given by the differential equation $$\frac{d\theta}{dt} = -k(\theta - \theta_0),$$ where \(\theta_0\)°F is the air temperature and \(k\) is a constant.

1. Show by integration that the solution of this equation is \(\theta = \theta_0 + Ae^{-kt}\), where \(A\) is a constant. [5]

The value of \(\theta_0\) is 50, and the initial value of \(\theta\) is 98. The initial rate of temperature loss is 1.5°F per hour.

1. Find \(A\), and show that \(k = 0.03125\). [4]
2. Use this model to calculate how long it will take for the temperature to drop
   1. from 98°F to 89°F,
   2. from 98°F to 80°F. [5]
3. Comment on the results obtained in parts (i) and (iv). [1]

Charlie buys a car for £18000 on 1 January 2016. The value of the car decreases exponentially. The car has a value of £12000 on 1 January 2018.

1. Charlie says: • because the car has lost £6000 after two years, after another two years it will be worth £6000. Charlie's friend Kaya says: • because the car has lost one third of its value after two years, after another two years it will be worth £8000. Explain whose statement is correct, justifying the value they have stated. [2 marks]
2. The value of Charlie's car, £\(V\), \(t\) years after 1 January 2016 may be modelled by the equation $$V = Ae^{-kt}$$ where \(A\) and \(k\) are positive constants. Find the value of \(t\) when the car has a value of £10000, giving your answer to two significant figures. [5 marks]
3. Give a reason why the model, in this context, will not be suitable to calculate the value of the car when \(t = 30\) [1 mark]

Trees in a forest may be affected by one of two types of fungal disease, but not by both. The number of trees affected by disease A, \(n_A\), can be modelled by the formula $$n_A = ae^{0.1t}$$ where \(t\) is the time in years after 1 January 2017. The number of trees affected by disease B, \(n_B\), can be modelled by the formula $$n_B = be^{0.2t}$$ On 1 January 2017 a total of 290 trees were affected by a fungal disease. On 1 January 2018 a total of 331 trees were affected by a fungal disease.

1. Show that \(b = 90\), to the nearest integer, and find the value of \(a\). [3 marks]
2. Estimate the total number of trees that will be affected by a fungal disease on 1 January 2020. [1 mark]
3. Find the year in which the number of trees affected by disease B will first exceed the number affected by disease A. [3 marks]
4. Comment on the long-term accuracy of the model. [1 mark]

The population of a country was 3.6 million in 1989. It grew exponentially to reach 6 million in 2019. Estimate the population of the country in 2049 if the exponential growth continues unchanged. [2 marks]

A singer has a social media account with a number of followers. The singer releases a new song and the number of followers grows exponentially. The number of followers, \(F\), may be modelled by the formula $$F = ae^{kt}$$ where \(t\) is the number of days since the song was released and \(a\) and \(k\) are constants. • Two days after the song is released the account has 2050 followers. • Five days after the song is released the account has 9200 followers. On the graph below ln \(F\) has been plotted against \(t\) for these two pieces of data. A line has been drawn passing through these two data points. \includegraphics{figure_2}

1. 1. Show that \(\ln F = \ln a + kt\) [2 marks]
   2. Using the graph, estimate the value of the constant \(a\) and the value of the constant \(k\) [4 marks]
2. 1. Show that \(\frac{dF}{dt} = kF\) [2 marks]
   2. Using the model, estimate the rate at which the number of followers is increasing 5 days after the song is released. [2 marks]
3. The singer claims that 30 days after the song is released, the account will have more than a billion followers. Comment on the singer's claim. [1 mark]

David has been investigating the population of rabbits on an island during a three-year period. Based on data that he has collected, David decides to model the population of rabbits, \(R\), by the formula $$R = 50e^{0.5t}$$ where \(t\) is the time in years after 1 January 2016.

1. Using David's model:
   1. state the population of rabbits on the island on 1 January 2016; [1 mark]
   2. predict the population of rabbits on 1 January 2021. [1 mark]
2. Use David's model to find the value of \(t\) when \(R = 150\), giving your answer to three significant figures. [2 marks]
3. Give one reason why David's model may not be appropriate. [1 mark]
4. On the same island, the population of crickets, \(C\), can be modelled by the formula $$C = 1000e^{-0.1t}$$ where \(t\) is the time in years after 1 January 2016. Using the two models, find the year during which the population of rabbits first exceeds the population of crickets. [3 marks]

Theresa bought a house on 2 January 1970 for £8000. The house was valued by a local estate agent on the same date every 10 years up to 2010. The valuations are shown in the following table.

Year	1970	1980	1990	2000	2010
Valuation price	£8000	£19000	£36000	£82000	£205000

The valuation price of the house can be modelled by the equation $$V = pq^t$$ where \(V\) pounds is the valuation price \(t\) years after 2 January 1970 and \(p\) and \(q\) are constants.

1. Show that \(V = pq^t\) can be written as \(\log_{10} V = \log_{10} p + t \log_{10} q\) [2 marks]
2. The values in the table of \(\log_{10} V\) against \(t\) have been plotted and a line of best fit has been drawn on the graph below. \includegraphics{figure_8b} Using the given line of best fit, find estimates for the values of \(p\) and \(q\). Give your answers correct to three significant figures. [4 marks]
3. Determine the year in which Theresa's house will first be worth half a million pounds. [3 marks]
4. Explain whether your answer to part (c) is likely to be reliable. [2 marks]

A zoologist is investigating the growth of a population of red squirrels in a forest. She uses the equation \(N = \frac{200}{1 + 9e^{-\frac{t}{5}}}\) as a model to predict the number of squirrels, \(N\), in the population \(t\) weeks after the start of the investigation. What is the size of the squirrel population at the start of the investigation? Circle your answer. [1 mark] \(5\) \(\quad\) \(20\) \(\quad\) \(40\) \(\quad\) \(200\)

A student is conducting an experiment in a laboratory to investigate how quickly liquids cool to room temperature. A beaker containing a hot liquid at an initial temperature of \(75°C\) cools so that the temperature, \(\theta °C\), of the liquid at time \(t\) minutes can be modelled by the equation $$\theta = 5(4 + \lambda e^{-kt})$$ where \(\lambda\) and \(k\) are constants. After 2 minutes the temperature falls to \(68°C\).

1. Find the temperature of the liquid after 15 minutes. Give your answer to three significant figures. [7 marks]
2. 1. Find the room temperature of the laboratory, giving a reason for your answer. [2 marks]
   2. Find the time taken in minutes for the liquid to cool to \(1°C\) above the room temperature of the laboratory. [2 marks]
3. Explain why the model might need to be changed if the experiment was conducted in a different place. [1 mark]

The number of radioactive atoms, \(N\), in a sample of a sodium isotope after time \(t\) hours can be modelled by $$N = N_0 e^{-kt}$$ where \(N_0\) is the initial number of radioactive atoms in the sample and \(k\) is a positive constant. The model remains valid for large numbers of atoms.

1. It takes 15.9 hours for half of the sodium atoms to decay. Determine the number of days required for at least 90\% of the number of atoms in the original sample to decay. [5 marks]
2. Find the percentage of the atoms remaining after the first week. Give your answer to two significant figures. [2 marks]
3. Explain why the model can only provide an estimate for the number of remaining atoms. [1 mark]
4. Explain why the model is invalid in the long run. [1 mark]