1.06i Exponential growth/decay: in modelling context

\includegraphics{figure_4} The value of a sculpture, \(£V\), is modelled by the equation \(V = Ap^t\), where \(A\) and \(p\) are constants and \(t\) is the number of years since the value of the painting was first recorded on 1st January 1960. The line \(l\) shown in Figure 4 illustrates the linear relationship between \(t\) and \(\log_{10}V\) for \(t \geq 0\). The line \(l\) passes through the point \((0, \log_{10}20)\) and \((50, \log_{10}2000)\).

1. Write down the equation of the line \(l\). [3]
2. Using your answer to part a or otherwise, find the values of \(A\) and \(p\). [4]
3. With reference to the model, interpret the values of the constant \(A\) and \(p\). [2]
4. Use your model, to predict the value of the sculpture, on 1st January 2020, giving your answer to the nearest pounds. [1]

The size \(N\) of the population of a small island at time \(t\) years may be modelled by \(N = Ae^{kt}\), where \(A\) and \(k\) are constants. It is known that \(N = 100\) when \(t = 2\) and that \(N = 160\) when \(t = 12\).

1. Interpret the constant \(A\) in the context of the question. [1]
2. Show that \(k = 0.047\), correct to three decimal places. [4]
3. Find the size of the population when \(t = 20\). [3]

\includegraphics{figure_2} A town's population, \(P\), is modelled by the equation \(P = ab^t\), where \(a\) and \(b\) are constants and \(t\) is the number of years since the population was first recorded. The line \(l\) shown in Figure 2 illustrates the linear relationship between \(t\) and \(\log_{10} P\) for the population over a period of 100 years. The line \(l\) meets the vertical axis at \((0, 5)\) as shown. The gradient of \(l\) is \(\frac{1}{200}\).

1. Write down an equation for \(l\). [2]
2. Find the value of \(a\) and the value of \(b\). [4]
3. With reference to the model interpret
   1. the value of the constant \(a\),
   2. the value of the constant \(b\).
   [2]
4. Find
   1. the population predicted by the model when \(t = 100\), giving your answer to the nearest hundred thousand,
   2. the number of years it takes the population to reach 200000, according to the model.
   [3]
5. State two reasons why this may not be a realistic population model. [2]

A scientist is studying the growth of two different populations of bacteria. The number of bacteria, \(N\), in the first population is modelled by the equation $$N = Ae^{kt} \quad t \geq 0$$ where \(A\) and \(k\) are positive constants and \(t\) is the time in hours from the start of the study. Given that • there were 1000 bacteria in this population at the start of the study • it took exactly 5 hours from the start of the study for this population to double

1. find a complete equation for the model. [4]
2. Hence find the rate of increase in the number of bacteria in this population exactly 8 hours from the start of the study. Give your answer to 2 significant figures. [2]

The number of bacteria, \(M\), in the second population is modelled by the equation $$M = 500e^{1.4t} \quad t \geq 0$$ where \(k\) has the value found in part (a) and \(t\) is the time in hours from the start of the study. Given that \(T\) hours after the start of the study, the number of bacteria in the two different populations was the same,

1. find the value of \(T\). [3]

A treatment is used to reduce the concentration of nitrate in the water in a pond. The concentration of nitrate in the pond water, \(N\) ppm (parts per million), is modelled by the equation $$N = 65 - 3e^{0.1t} \quad t \in \mathbb{R} \quad t \geq 0$$ where \(t\) hours is the time after the treatment was applied. Use the equation of the model to answer parts (a) and (b).

1. Calculate the reduction in the concentration of nitrate in the pond water in the first 8 hours after the treatment was applied. [3] For fish to survive in the pond, the concentration of nitrate in the water must be no more than 20 ppm.
2. Calculate the minimum time, after the treatment is applied, before fish can be safely introduced into the pond. Give your answer in hours to one decimal place. [3]

In a chemical reaction, the mass \(m\) grams of a chemical after \(t\) minutes is modelled by the equation $$m = 20 + 30e^{-0.1t}.$$

1. Find the initial mass of the chemical. What is the mass of chemical in the long term? [3]
2. Find the time when the mass is 30 grams. [3]
3. Sketch the graph of \(m\) against \(t\). [2]

The mass of a substance is decreasing exponentially. Its mass is \(m\) grams at time \(t\) years. The following table shows certain values of \(t\) and \(m\).

1. Find the values missing from the table. [2]
2. Determine the value of \(t\), correct to the nearest integer, for which the mass is 50 grams. [4]

Helga invests £4000 in a savings account. After \(t\) days, her investment is worth \(£y\). The rate of increase of \(y\) is \(ky\), where \(k\) is a constant.

1. Write down a differential equation in terms of \(t\), \(y\) and \(k\). [1]
2. Solve your differential equation to find the value of Helga's investment after \(t\) days. Give your answer in terms of \(k\) and \(t\). [4]

It is given that \(k = \frac{r}{365}\ln\left(1 + \frac{r}{100}\right)\) where \(r\%\) is the rate of interest per annum. During the first year the rate of interest is 6% per annum.

1. Find the value of Helga's investment after 90 days. [2]

After one year (365 days), the rate of interest drops to 5% per annum.

1. Find the total time that it will take for Helga's investment to double in value. [5]

The number of members of a social networking site is modelled by \(m = 150e^{2t}\), where \(m\) is the number of members and \(t\) is time in weeks after the launch of the site.

1. State what this model implies about the relationship between \(m\) and the rate of change of \(m\). [2]
2. What is the significance of the integer 150 in the model? [1]
3. Find the week in which the model predicts that the number of members first exceeds 60 000. [3]
4. The social networking site only expects to attract 60 000 members. Suggest how the model could be refined to take account of this. [1]

Diane is given an injection that combines two drugs, Antiflu and Coldcure. At time \(t\) hours after the injection, the concentration of Antiflu in Diane's bloodstream is \(3e^{-0.02t}\) units and the concentration of Coldcure is \(5e^{-0.07t}\) units. Each drug becomes ineffective when its concentration falls below 1 unit.

1. Show that Coldcure becomes ineffective before Antiflu. [3]
2. Sketch, on the same diagram, the graphs of concentration against time for each drug. [2]
3. 20 hours after the first injection, Diane is given a second injection. Determine the concentration of Coldcure 10 hours later. [2]