Linear transformation to find constants

12 Answer part (ii) of this question on the insert provided. Since 1945 the populations of many countries have been growing. The table shows the estimated population of 15- to 59-year-olds in Africa during the period 1955 to 2005. Source: United Nations Such estimates are used to model future population growth and world needs of resources. One model is \(P = a 10 ^ { b t }\), where the population is \(P\) millions, \(t\) is the number of years after 1945 and \(a\) and \(b\) are constants.

1. Show that, using this model, the graph of \(\log _ { 10 } P\) against \(t\) is a straight line of gradient \(b\). State the intercept of this line on the vertical axis.
2. On the insert, complete the table, giving values correct to 2 decimal places, and plot the graph of \(\log _ { 10 } P\) against \(t\). Draw, by eye, a line of best fit on your graph.
3. Use your graph to find the equation for \(P\) in terms of \(t\).
4. Use your results to estimate the population of 15- to 59-year-olds in Africa in 2050. Comment, with a reason, on the reliability of this estimate.

12 Answer the whole of this question on the insert provided. A colony of bats is increasing. The population, \(P\), is modelled by \(P = a \times 10 ^ { b t }\), where \(t\) is the time in years after 2000.

1. Show that, according to this model, the graph of \(\log _ { 10 } P\) against \(t\) should be a straight line of gradient \(b\). State, in terms of \(a\), the intercept on the vertical axis.
2. The table gives the data for the population from 2001 to 2005.

   Year	2001	2002	2003	2004	2005
   \(t\)	1	2	3	4	5
   \(P\)	7900	8800	10000	11300	12800

   Complete the table of values on the insert, and plot \(\log _ { 10 } P\) against \(t\). Draw a line of best fit for the data.
3. Use your graph to find the equation for \(P\) in terms of \(t\).
4. Predict the population in 2008 according to this model.

9 Answer parts (ii) and (iii) of this question on the Insert provided. The bat population of a colony is being investigated and data are collected of the estimated number of bats in the colony at the beginning of each year. It is thought that the population may be modelled by the formula $$P = P _ { 0 } \mathrm { e } ^ { k t }$$ where \(P _ { 0 }\) and \(k\) are constants, \(P\) is the number of bats and \(t\) is the number of years after the start of the collection of data.

1. Explain why a graph of \(\ln P\) against \(t\) should give a straight line. State the gradient and intercept of this line.
2. The data collected are as follows.

   Time \(( t\) years \()\)	0	1	2	3	4
   Number of bats, \(P\)	100	170	300	340	360

   Using the first three pairs of data in the table, plot \(\ln P\) against \(t\) on the axes given on the Insert, and hence estimate values for \(P _ { 0 }\) and \(k\).
   (Work to three significant figures.) This model assumes exponential growth, and assumes that once born a bat does not die, continuing to reproduce. This is unrealistic and so a second model is proposed with formula $$P = 150 \arctan ( t - 1 ) + 170$$ (You are reminded that arctan values should be given in radians.)
3. Plot on a single graph on the Insert the curves \(P = P _ { 0 } \mathrm { e } ^ { k t }\) for your values of \(P _ { 0 }\) and \(k\) and \(P = 150 \arctan ( t - 1 ) + 170\). The data pairs in the table above have been plotted for you.
4. Using the second model calculate an estimate of the number of years it is before the bat population exceeds 375. \section*{Insert for question 3.}
5. Sketch the graph of \(y = 2 \mathrm { f } ( x )\) \includegraphics[max width=\textwidth, alt={}, center]{3853d1e7-ae1f-4eca-93c7-96f03b6d31c3-6_641_1431_541_354}
6. Sketch the graph of \(y = \mathrm { f } ( 2 x )\). \includegraphics[max width=\textwidth, alt={}, center]{3853d1e7-ae1f-4eca-93c7-96f03b6d31c3-6_691_1539_1468_374} \section*{Insert for question 9.}
7. Plot \(\ln P\) against \(t\). \includegraphics[max width=\textwidth, alt={}, center]{3853d1e7-ae1f-4eca-93c7-96f03b6d31c3-7_704_1442_443_338}
8. Plot the curves \(P = P _ { 0 } \mathrm { e } ^ { k t }\) and \(P = 150 \arctan ( t - 1 ) + 170\) for your values of \(P _ { 0 }\) and \(k\). The data pairs are plotted on the graph. \includegraphics[max width=\textwidth, alt={}, center]{3853d1e7-ae1f-4eca-93c7-96f03b6d31c3-7_780_1399_1546_333}

9 The points \(( 2,6 )\) and \(( 3,18 )\) lie on the curve \(y = a x ^ { n }\).
Use logarithms to find the values of \(a\) and \(n\), giving your answers correct to 2 decimal places.

7 Scientists observed a colony of seabirds over a period of 10 years starting in 2010. They concluded that the number of birds in the colony, its population \(P\), could be modelled by a formula of the form $$P = a \left( 10 ^ { b t } \right)$$ where \(t\) is the time in years after 2010, and \(a\) and \(b\) are constants.
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1. Explain what the value of \(a\) represents.
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2. Show that \(\log _ { 10 } P = b t + \log _ { 10 } a\) 7
3. The table below contains some data collected by the scientists.

   Year	2013	2015
   \(t\)	3
   \(P\)	10200	12800
   \(\log _ { 10 } P\)	4.0086

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4. 1. Complete the table, giving the \(\log _ { 10 } P\) value to 5 significant figures.
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5. (ii) Use the data to calculate the value of \(a\) and the value of \(b\).
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6. (iii) Use the model to estimate the population of the colony in 2024.
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7. 1. State an assumption that must be made in using the model to estimate the population of the colony in 2024.
      [0pt] [1 mark] 7
8. (ii) Hence comment, with a reason, on the reliability of your estimate made in part (c)(iii).
   [0pt] [1 mark] \includegraphics[max width=\textwidth, alt={}, center]{1f887565-4587-4520-99d4-f3635b015525-11_2488_1730_219_141}

10 As part of an experiment, Zena puts a bucket of hot water outside on a day when the outside temperature is \(0 ^ { \circ } \mathrm { C }\). She measures the temperature of the water after 10 minutes and after 20 minutes. Her results are shown below. Zena models the relationship between \(\theta\), the temperature of the water in \({ } ^ { \circ } \mathrm { C }\), and \(t\), the time in minutes, by $$\theta = A \times 10 ^ { - k t }$$ where \(A\) and \(k\) are constants. 10

1. Using \(t = 0\), explain how the value of \(A\) relates to the experiment. 10
2. Show that $$\log _ { 10 } \theta = \log _ { 10 } A - k t$$ 10
3. Using Zena's results, calculate the values of \(A\) and \(k\).
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4. Zena states that the temperature of the water will be less than \(1 ^ { \circ } \mathrm { C }\) after 45 minutes. Determine whether the model supports this statement.
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5. Explain why Zena's model is unlikely to accurately give the value of \(\theta\) after 45 minutes.

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1. (ii) Using the graph, estimate the value of the constant \(a\) and the value of the constant \(k\) [4 marks]
   \hline \end{tabular} \end{center} 10
2. 1. Show that \(\frac { \mathrm { d } F } { \mathrm {~d} t } = k F\) 10
3. (ii) Using the model, estimate the rate at which the number of followers is increasing 5 days after the song is released.
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4. 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.

6 Victoria, a market researcher, believes the average weekly value, \(\pounds V\) million, of online grocery sales in the UK has grown exponentially since 2009. Victoria models the incomplete data, shown in the table, using the formula $$V = a \times b ^ { N }$$ where \(N\) is the number of years since 2009 and \(a\) and \(b\) are constants. 6

1. Victoria wishes to determine the values of \(a\) and \(b\) in her formula.
   To do this she plots a graph of \(\log _ { 10 } V\) against \(N\) and then draws a line of best fit as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{de8a7d38-a665-4feb-854e-ac83f413d133-08_757_1040_1169_589} The equation of Victoria's line of best fit is $$\log _ { 10 } V = 0.057 N + 1.76$$ 6
2. 1. Use the equation of Victoria's line of best fit to show that, correct to three significant figures, \(a = 57.5\) [0pt] [1 mark]
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3. (ii) Use the equation of Victoria's line of best fit to find the value of \(b\) Give your answer to three significant figures. 6
4. According to Victoria's model, state the yearly percentage increase in the average weekly value of online grocery sales. 6
5. 1. Use Victoria's model to predict the average weekly value of online grocery sales in 2025.
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6. (ii) Explain why the prediction made in part (c)(i) may be unreliable.