1.06h Logarithmic graphs: reduce y=ax^n and y=kb^x to linear form

8. The owner of a new company models the number of customers that the company will have at the end of each month. The owner assumes that

• a constant proportion, \(p\) (where \(0 < p < 1\) ), of the previous month's customers will be retained for the next month
• a constant number of new customers, \(k\), will be added each month.

Let \(u _ { n }\) (where \(n \geqslant 1\) ) represent the number of customers that the company will have at the end of \(n\) months. The company has 5000 customers at the end of the first month.

1. By setting up a first order recurrence relation for \(u _ { n + 1 }\) in terms of \(u _ { n }\), determine an expression for \(u _ { n }\) in terms of \(n , p\) and \(k\). The owner believes that \(95 \%\) of the previous month's customers will be retained each month and that there will be 10000 new customers each month. According to the model, the company will first have at least 135000 customers by the end of the \(m\) th month.
2. Using logarithms, determine the value of \(m\). Please check the examination details below before entering your candidate information \section*{Further Mathematics} Advanced
   PAPER 4D: Decision Mathematics 2 \section*{Answer Book} Do not return the question paper with the answer book.
   1.

   	1	2	3	4
   A	32	45	34	48
   B	37	39	50	46
   C	46	44	40	42
   D	43	45	48	52

   You may not need to use all of these tables.

   	1	2	3	4
   A
   B
   C
   D

   	1	2	3	4
   A
   B
   C
   D

   	1	2	3	4
   A
   B
   C
   D

   	1	2	3	4
   A
   B
   C
   D

   VALV SIHI NI IIIIIM ION OC
   VIAV SIUL NI JAIIM ION OC
   VIAV SIHI NI III IM ION OC

   	1	2	3	4
   A
   B
   C
   D

   	1	2	3	4
   \(A\)
   \(B\)
   \(C\)
   \(D\)

   	1	2	3	4
   A
   B
   C
   D

   	1	2	3	4
   A
   B
   C
   D

   	1	2	3	4
   A
   B
   C
   D

   2.

   VAMV SIHI NI IIIHM ION OO

   VIAV SIHI NI JIIIM I ON OC

   VJYV SIHI NI JIIIM ION OC

   3. 4. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{cea07472-f93b-4a7b-b362-89fb8c0af4a9-16_936_1317_255_376} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure} 5.

3. 	\(R\)	\(S\)	\(T\)
   \(A\)
   \(B\)
   \(C\)
   \(D\)

   	\(R\)	\(S\)	\(T\)
   \(A\)
   \(B\)
   \(C\)
   \(D\)

   	\(R\)	\(S\)	\(T\)
   \(A\)
   \(B\)
   \(C\)
   \(D\)

   	\(R\)	\(S\)	\(T\)
   \(A\)
   \(B\)
   \(C\)
   \(D\)

   	\(R\)	\(S\)	\(T\)
   \(A\)
   \(B\)
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   \(D\)

   6.

   Stage	State	Action	Destination	Value
   May	2	0	0	160
   	1	1	0	80 + 35
   	0	2	0	70

   Stage	State	Action	Destination	Value

   Month	January	February	March	April	May
   Number made

   Minimum cost: \(\_\_\_\_\) 7. Player A \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{Player B}

   	Option W	Option X	Option Y	Option Z
   Option Q	4	3	-1	-2
   Option R	-3	5	-4	\(k\)
   Option S	-1	6	3	-3

   \end{table} The tableau for (d)(ii) can be found at the bottom of page 16

   b.v.				□	□	□	□		Value
   \includegraphics[max width=\textwidth, alt={}]{cea07472-f93b-4a7b-b362-89fb8c0af4a9-24_56_77_2348_182}		□	□
   P

   8.

6 The table below gives the population of breeding pairs of red kites in Yorkshire from 2001 to 2008. Source: \href{http://www.gigrin.co.uk}{www.gigrin.co.uk}
The following model for the population has been proposed: $$N = a \times b ^ { t } ,$$ where \(N\) is the number of breeding pairs \(t\) years after the year 2000, and \(a\) and \(b\) are constants.

1. Show that the model can be transformed to a linear relationship between \(\log _ { 10 } N\) and \(t\).
2. On graph paper, plot \(\log _ { 10 } N\) against \(t\) and draw by eye a line of best fit. Use your line to estimate the values of \(a\) and \(b\) in the equation for \(N\) in terms of \(t\).
3. What values of \(N\) does the model give for the years 2008 and 2020?
4. In which year will the number of breeding pairs first exceed 500 according to the model?
5. Comment on the suitability of the model to predict the population of breeding pairs of red kites in Yorkshire.

4 A population, \(P\), is modelled by the equation \(P = a \mathrm { e } ^ { b t }\) where \(t\) is time in years, and \(a\) and \(b\) are constants.

1. By considering logarithms, show that a graph of \(\ln P\) against \(t\) is a straight line. State the intercept on the vertical axis and the gradient.
2. Use the graph below to obtain values for \(a\) and \(b\). \includegraphics[max width=\textwidth, alt={}, center]{816a16df-e3a5-48ae-84c6-7f6f5bbba9ca-2_657_750_1530_740}

\includegraphics{figure_2} Two variable quantities \(x\) and \(y\) are related by the equation $$y = k(a^{-x}),$$ where \(a\) and \(k\) are constants. Four pairs of values of \(x\) and \(y\) are measured experimentally. The result of plotting \(\ln y\) against \(x\) is shown in the diagram. Use the diagram to estimate the values of \(a\) and \(k\). [5]

\includegraphics{figure_2} The variables \(x\) and \(y\) satisfy the equation \(y = Ae^{px}\), where \(A\) and \(p\) are constants. The graph of \(\ln y\) against \(x\) is a straight line passing through the points \((5, 3.17)\) and \((10, 4.77)\), as shown in the diagram. Find the values of \(A\) and \(p\) correct to 2 decimal places. [5]

\includegraphics{figure_2} The variables \(x\) and \(y\) satisfy the equation \(y = Kx^p\), where \(K\) and \(p\) are constants. The graph of \(\ln y\) against \(\ln x\) is a straight line passing through the points \((1.28, 3.69)\) and \((2.11, 4.81)\), as shown in the diagram. Find the values of \(K\) and \(p\) correct to 2 decimal places. [5]

\includegraphics{figure_4} The variables \(x\) and \(y\) satisfy the equation \(ky = e^{cx}\), where \(k\) and \(c\) are constants. The graph of \(\ln y\) against \(x\) is a straight line passing through the points \((2.80, 0.372)\) and \((5.10, 2.21)\), as shown in the diagram. Find the values of \(k\) and \(c\). Give each value correct to 2 significant figures. [4]

\includegraphics{figure_6} The variables \(x\) and \(y\) satisfy the equation \(ay = b^x\), where \(a\) and \(b\) are constants. The graph of \(\ln y\) against \(x\) is a straight line passing through the points \((0.50, 2.24)\) and \((3.40, 8.27)\), as shown in the diagram. Find the values of \(a\) and \(b\). Give each value correct to 1 significant figure. [4]

The variables \(x\) and \(y\) satisfy the equation \(y^3 = Ae^{2x}\), where \(A\) is a constant. The graph of \(\ln y\) against \(x\) is a straight line.

1. Find the gradient of this line. [2]
2. Given that the line intersects the axis of \(\ln y\) at the point where \(\ln y = 0.5\), find the value of \(A\) correct to 2 decimal places. [2]

\includegraphics{figure_3} The variables \(x\) and \(y\) satisfy the equation \(y = Ae^{-kx^2}\), where \(A\) and \(k\) are constants. The graph of \(\ln y\) against \(x^2\) is a straight line passing through the points \((0.64, 0.76)\) and \((1.69, 0.32)\), as shown in the diagram. Find the values of \(A\) and \(k\) correct to 2 decimal places. [5]

Two variable quantities \(x\) and \(y\) are believed to satisfy an equation of the form \(y = C(a^x)\), where \(C\) and \(a\) are constants. An experiment produced four pairs of values of \(x\) and \(y\). The table below gives the corresponding values of \(x\) and \(\ln y\). By plotting \(\ln y\) against \(x\) for these four pairs of values and drawing a suitable straight line, estimate the values of \(C\) and \(a\). Give your answers correct to 2 significant figures. [5] \includegraphics{figure_2}

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 \(P = a10^{bt}\), 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. [3]
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]
3. Use your graph to find the equation for \(P\) in terms of \(t\). [4]
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. [3]

The table shows population data for a country. The data may be represented by an exponential model of growth. Using \(t\) as the number of years after 1960, a suitable model is \(p = a \times 10^{kt}\).

1. Derive an equation for \(\log_{10} p\) in terms of \(a\), \(k\) and \(t\). [2]
2. Complete the table and draw the graph of \(\log_{10} p\) against \(t\), drawing a line of best fit by eye. [3]
3. Use your line of best fit to express \(\log_{10} p\) in terms of \(t\) and hence find \(p\) in terms of \(t\). [4]
4. According to the model, what was the population in 1960? [1]
5. According to the model, when will the population reach 200 million? [3]

Answer the whole of this question on the insert provided. A colony of bats is increasing. The population, \(P\), is modelled by \(P = a \times 10^{bt}\), 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. [3]
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]
3. Use your graph to find the equation for \(P\) in terms of \(t\). [4]
4. Predict the population in 2008 according to this model. [2]

The percentage of the adult population visiting the cinema in Great Britain has tended to increase since the 1980s. The table shows the results of surveys in various years. Source: Department of National Statistics, www.statistics.gov.uk This growth may be modelled by an equation of the form $$P = at^b,$$ where \(P\) is the percentage of the adult population visiting the cinema, \(t\) is the number of years after the year 1985/86 and \(a\) and \(b\) are constants to be determined.

1. Show that, according to this model, the graph of \(\log_{10} P\) against \(\log_{10} t\) should be a straight line of gradient \(b\). State, in terms of \(a\), the intercept on the vertical axis. [3]
2. Complete the table of values on the insert, and plot \(\log_{10} P\) against \(\log_{10} t\). Draw by eye a line of best fit for the data. [4]
3. Use your graph to find the equation for \(P\) in terms of \(t\). [4]
4. Predict the percentage of the adult population visiting the cinema in the year 2007/2008 (i.e. when \(t = 22\)), according to this model. [1]

The points \((2, 6)\) and \((3, 18)\) lie on the curve \(y = ax^n\). Use logarithms to find the values of \(a\) and \(n\), giving your answers correct to 2 decimal places. [5]

A hot drink when first made has a temperature which is \(65°C\) higher than room temperature. The temperature difference, \(d °C\), between the drink and its surroundings decreases by \(1.7\%\) each minute.

1. Show that 3 minutes after the drink is made, \(d = 61.7\) to 3 significant figures. [2]
2. Write down an expression for the value of \(d\) at time \(n\) minutes after the drink is made, where \(n\) is an integer. [1]
3. Show that when \(d < 3\), \(n\) must satisfy the inequality $$n > \frac{\log_{10} 3 - \log_{10} 65}{\log_{10} 0.983}.$$ Hence find the least integer value of \(n\) for which \(d < 3\). [4]
4. The temperature difference at any time \(t\) minutes after the drink is made can also be expressed as \(d = 65 \times 10^{-kt}\), for some constant \(k\). Use the value of \(d\) for 1 minute after the drink is made to calculate the value of \(k\). Hence find the temperature difference 25.3 minutes after the drink is made. [4]

The thickness of a glacier has been measured every five years from 1960 to 2010. The table shows the reduction in thickness from its measurement in 1960. An exponential model may be used for these data, assuming that the relationship between \(h\) and \(t\) is of the form \(h = a \times 10^{bt}\), where \(a\) and \(b\) are constants to be determined.

1. Show that this relationship may be expressed in the form \(\log_{10} h = mt + c\), stating the values of \(m\) and \(c\) in terms of \(a\) and \(b\). [2]
2. Complete the table of values in the answer book, giving your answers correct to 2 decimal places, and plot the graph of \(\log_{10} h\) against \(t\), drawing by eye a line of best fit. [4]
3. Use your graph to find \(h\) in terms of \(t\) for this model. [4]
4. Calculate by how much the glacier will reduce in thickness between 2010 and 2020, according to the model. [2]
5. Give one reason why this model will not be suitable in the long term. [1]

There are many different flu viruses. The numbers of flu viruses detected in the first few weeks of the 2012–2013 flu epidemic in the UK were as follows.

Week	1	2	3	4	5	6	7	8	9	10
Number of flu viruses	7	10	24	32	40	38	63	96	234	480

These data may be modelled by an equation of the form \(y = a \times 10^{bt}\), where \(y\) is the number of flu viruses detected in week \(t\) of the epidemic, and \(a\) and \(b\) are constants to be determined.

1. Explain why this model leads to a straight-line graph of \(\log_{10} y\) against \(t\). State the gradient and intercept of this graph in terms of \(a\) and \(b\). [3]
2. Complete the values of \(\log_{10} y\) in the table, draw the graph of \(\log_{10} y\) against \(t\), and draw by eye a line of best fit for the data. Hence determine the values of \(a\) and \(b\) and the equation for \(y\) in terms of \(t\) for this model. [8]

During the decline of the epidemic, an appropriate model was $$y = 921 \times 10^{-0.137w},$$ where \(y\) is the number of flu viruses detected in week \(w\) of the decline.

1. Use this to find the number of viruses detected in week 4 of the decline. [1]

The variables \(y\) and \(x\) are related by an equation of the form $$y = a(b^x)$$ where \(a\) and \(b\) are positive constants. Let \(Y = \log_{10} y\).

1. Show that there is a linear relationship between \(Y\) and \(x\). [2 marks]
2. The graph of \(Y\) against \(x\), shown below, passes through the points \((0, 2.5)\) and \((5, 0.5)\). \includegraphics{figure_3}
   1. Find the gradient of the line. [1 mark]
   2. Find the value of \(a\) and the value of \(b\), giving each answer to three significant figures. [4 marks]

Maxine measures the pressure, \(P\) kilopascals, and the volume, \(V\) litres, in a fixed quantity of gas. Maxine believes that the pressure and volume are connected by the equation $$P = cV^d$$ where \(c\) and \(d\) are constants. Using four experimental results, Maxine plots \(\log_{10} P\) against \(\log_{10} V\), as shown in the graph below. \includegraphics{figure_8}

1. Find the value of \(P\) and the value of \(V\) for the data point labelled \(A\) on the graph. [2 marks]
2. Calculate the value of each of the constants \(c\) and \(d\). [4 marks]
3. Estimate the pressure of the gas when the volume is \(2\) litres. [2 marks]

Raj is investigating how the price, \(P\) pounds, of a brilliant-cut diamond ring is related to the weight, \(C\) carats, of the diamond. He believes that they are connected by a formula $$P = aC^n$$ where \(a\) and \(n\) are constants.

1. Express \(\ln P\) in terms of \(\ln C\). [2 marks]
2. Raj researches the price of three brilliant-cut diamond rings on a website with the following results.

   \(C\)	0.60	1.15	1.50
   \(P\)	495	1200	1720

   1. Plot \(\ln P\) against \(\ln C\) for the three rings on the grid below. [2 marks] \includegraphics{figure_10b}
   2. Explain which feature of the plot suggests that Raj's belief may be correct. [1 mark]
   3. Using the graph on page 15, estimate the value of \(a\) and the value of \(n\). [4 marks]
3. Explain the significance of \(a\) in this context. [1 mark]
4. Raj wants to buy a ring with a brilliant-cut diamond of weight 2 carats. Estimate the price of such a ring. [2 marks]

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 $$P = a(10^{bt})$$ where \(t\) is the time in years after 2010, and \(a\) and \(b\) are constants.

1. Explain what the value of \(a\) represents. [1 mark]
2. Show that \(\log_{10} P = bt + \log_{10} a\) [2 marks]
3. The table below contains some data collected by the scientists.

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

   1. Complete the table, giving the \(\log_{10} P\) value to 5 significant figures. [1 mark]
   2. Use the data to calculate the value of \(a\) and the value of \(b\). [4 marks]
   3. Use the model to estimate the population of the colony in 2024. [2 marks]
4. 1. State an assumption that must be made in using the model to estimate the population of the colony in 2024. [1 mark]
   2. Hence comment, with a reason, on the reliability of your estimate made in part (c)(iii). [1 mark]

A student conducts an experiment and records the following data for two variables, \(x\) and \(y\).

\(x\)	1	2	3	4	5	6
\(y\)	14	45	130	1100	1300	3400
\(\log_{10} y\)

The student is told that the relationship between \(x\) and \(y\) can be modelled by an equation of the form \(y = kb^x\)

1. Plot values of \(\log_{10} y\) against \(x\) on the grid below. [2 marks] \includegraphics{figure_10}
2. State, with a reason, which value of \(y\) is likely to have been recorded incorrectly. [1 mark]
3. By drawing an appropriate straight line, find the values of \(k\) and \(b\). [4 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]