Exponential modelling from log-linear

8. A dose of antibiotics is given to a patient. The amount of the antibiotic, \(x\) milligrams, in the patient's bloodstream \(t\) hours after the dose was given, is found to satisfy the equation $$\log _ { 10 } x = 2.74 - 0.079 t$$

1. Show that this equation can be written in the form $$x = p q ^ { - t }$$ where \(p\) and \(q\) are constants to be found. Give the value of \(p\) to the nearest whole number and the value of \(q\) to 2 significant figures.
2. With reference to the equation in part (a), interpret the value of the constant \(p\). When a different dose of the antibiotic is given to another patient, the values of \(x\) and \(t\) satisfy the equation $$x = 400 \times 1.4 ^ { - t }$$
3. Use calculus to find, to 2 significant figures, the value of \(\frac { \mathrm { d } x } { \mathrm {~d} t }\) when \(t = 5\)

3. \begin{figure}[h] \end{figure} The line \(l\) in Figure 1 shows a linear relationship between \(\log _ { 10 } y\) and \(x\).
The line passes through the points \(( 0,1.5 )\) and \(( - 4.8,0 )\) as shown.

1. Write down an equation for \(l\).
2. Hence, or otherwise, express \(y\) in the form \(k b ^ { x }\), giving the values of the constants \(k\) and \(b\) to 3 significant figures.

1. The amount of money raised for a charity is being monitored.

The total amount raised in the \(t\) months after monitoring began, \(\pounds D\), is modelled by the equation $$\log _ { 10 } D = 1.04 + 0.38 t$$

1. Write this equation in the form $$D = a b ^ { t }$$ where \(a\) and \(b\) are constants to be found. Give each value to 4 significant figures. When \(t = T\), the total amount of money raised is \(\pounds 45000\) According to the model,
2. find the value of \(T\), giving your answer to 3 significant figures. The charity aims to raise a total of \(\pounds 350000\) within the first 12 months of monitoring.
   According to the model,
3. determine whether or not the charity will achieve its aim.

4. \begin{figure}[h] \end{figure} The number of subscribers to an online video streaming service, \(N\), is modelled by the equation $$N = a b ^ { t }$$ where \(a\) and \(b\) are constants and \(t\) is the number of years since monitoring began.
The line in Figure 1 shows the linear relationship between \(t\) and \(\log _ { 10 } N\) The line passes through the points \(( 0,3.08 )\) and \(( 5,3.85 )\) Using this information,

1. find an equation for this line.
2. Find the value of \(a\) and the value of \(b\), giving your answers to 3 significant figures. When \(t = T\) the number of subscribers is 500000 According to the model,
3. find the value of \(T\)

1. A scientist monitored the growth of bacteria on a dish over a 30 -day period.

The area, \(N \mathrm {~mm} ^ { 2 }\), of the dish covered by bacteria, \(t\) days after monitoring began, is modelled by the equation $$\log _ { 10 } N = 0.0646 t + 1.478 \quad 0 \leqslant t \leqslant 30$$

1. Show that this equation may be written in the form $$N = a b ^ { t }$$ where \(a\) and \(b\) are constants to be found. Give the value of \(a\) to the nearest integer and give the value of \(b\) to 3 significant figures.
2. Use the model to find the area of the dish covered by bacteria 30 days after monitoring began. Give your answer, in \(\mathrm { mm } ^ { 2 }\), to 2 significant figures.

7 The mass, \(M \mathrm {~kg}\), of a species of tree can be modelled by the equation $$\log _ { 10 } M = 1.93 \log _ { 10 } r + 0.684$$ where \(r \mathrm {~cm}\) is the base radius of the tree.
The base radius of a particular tree of this species is 45 cm .
According to the model,

1. find the mass of this tree, giving your answer to 2 significant figures.
2. Show that the equation of the model can be written in the form $$M = p r ^ { q }$$ giving the values of the constants \(p\) and \(q\) to 3 significant figures.
3. With reference to the model, interpret the value of the constant \(p\). Q

1. An area of sea floor is being monitored.

The area of the sea floor, \(S \mathrm {~km} ^ { 2 }\), covered by coral reefs is modelled by the equation $$S = p q ^ { t }$$ where \(p\) and \(q\) are constants and \(t\) is the number of years after monitoring began.
Given that $$\log _ { 10 } S = 4.5 - 0.006 t$$

1. find, according to the model, the area of sea floor covered by coral reefs when \(t = 2\)
2. find a complete equation for the model in the form $$S = p q ^ { t }$$ giving the value of \(p\) and the value of \(q\) each to 3 significant figures.
3. With reference to the model, interpret the value of the constant \(q\)

9 \begin{figure}[h] \end{figure} The graph of \(\log _ { 10 } y\) against \(x\) is a straight line as shown in Fig. 9 .

1. Find the equation for \(\log _ { 10 } y\) in terms of \(x\).
2. Find the equation for \(y\) in terms of \(x\). Section B (36 marks)

9 You are given that \(\log _ { 10 } y = 3 x + 2\).

1. Find the value of \(x\) when \(y = 500\), giving your answer correct to 2 decimal places.
2. Find the value of \(y\) when \(x = - 1\).
3. Express \(\log _ { 10 } \left( y ^ { 4 } \right)\) in terms of \(x\).
4. Find an expression for \(y\) in terms of \(x\). Section B (36 marks)

6 A mobile phone company records their annual sales on \(31 ^ { \text {st } }\) December every year.
Paul thinks that the annual sales, \(S\) million, can be modelled by the equation \(S = a b ^ { t }\), where \(a\) and \(b\) are both positive constants and \(t\) is the number of years since \(31 ^ { \text {st } }\) December 2015. Paul tests his theory by using the annual sales figures from \(31 ^ { \text {st } }\) December 2015 to \(31 { } ^ { \text {st } }\) December 2019. He plots these results on a graph, with \(t\) on the horizontal axis and \(\log _ { 10 } S\) on the vertical axis.

1. Explain why, if Paul's model is correct, the results should lie on a straight line of best fit on his graph. The results lie on a straight line of best fit which has a gradient of 0.146 and an intercept on the vertical axis of 0.583 .
2. Use these values to obtain estimates for \(a\) and \(b\), correct to 2 significant figures.
3. Use this model to predict the year in which, on the \(31 { } ^ { \text {st } }\) December, the annual sales would first be recorded as greater than 200 million.
4. Give a reason why this prediction may not be reliable.

13. \begin{figure}[h] \end{figure} The value of a rare painting, \(\pounds V\), is modelled by the equation \(V = p q ^ { t }\), where \(p\) and \(q\) are constants and \(t\) is the number of years since the value of the painting was first recorded on 1st January 1980. The line \(l\) shown in Figure 3 illustrates the linear relationship between \(t\) and \(\log _ { 10 } V\) since 1st January 1980. The equation of line \(l\) is \(\log _ { 10 } V = 0.05 t + 4.8\)

1. Find, to 4 significant figures, the value of \(p\) and the value of \(q\).
2. With reference to the model interpret
   1. the value of the constant \(p\),
   2. the value of the constant \(q\).
3. Find the value of the painting, as predicted by the model, on 1st January 2010, giving your answer to the nearest hundred thousand pounds.

1. An advertising agency is monitoring the number of views of an online advert.

The equation $$\log _ { 10 } V = 0.072 t + 2.379 \quad 1 \leqslant t \leqslant 30 , t \in \mathbb { N }$$ is used to model the total number of views of the advert, \(V\), in the first \(t\) days after the advert went live.

1. Show that \(V = a b ^ { t }\) where \(a\) and \(b\) are constants to be found. Give the value of \(a\) to the nearest whole number and give the value of \(b\) to 3 significant figures.
2. Interpret, with reference to the model, the value of \(a b\). Using this model, calculate
3. the total number of views of the advert in the first 20 days after the advert went live. Give your answer to 2 significant figures.

13. \begin{figure}[h] \end{figure} The value of a sculpture, \(\pounds V\), is modelled by the equation \(V = A p ^ { t }\), where \(A\) and \(p\) are constants and \(t\) is the number of years since the value of the painting was first recorded on \(1 ^ { \text {st } }\) 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 \(\left( 0 , \log _ { 10 } 20 \right)\) and \(\left( 50 , \log _ { 10 } 2000 \right)\).
a. Write down the equation of the line \(l\).
b. Using your answer to part a or otherwise, find the values of \(A\) and \(p\).
c. With reference to the model, interpret the values of the constant \(A\) and \(p\).
d. Use your model, to predict the value of the sculpture, on \(1 { } ^ { \text {st } }\) January 2020, giving your answer to the nearest pounds.

11. \begin{figure}[h] \end{figure} The value, \(V\) pounds, of a mobile phone, \(t\) months after it was bought, is modelled by $$V = a b ^ { t }$$ where \(a\) and \(b\) are constants.
Figure 2 shows the linear relationship between \(\log _ { 10 } V\) and \(t\).
The line passes through the points \(( 0,3 )\) and \(( 10,2.79 )\) as shown.
Using these points,

1. find the initial value of the phone,
2. find a complete equation for \(V\) in terms of \(t\), giving the exact value of \(a\) and giving the value of \(b\) to 3 significant figures. Exactly 2 years after it was bought, the value of the phone was \(\pounds 320\)
3. Use this information to evaluate the reliability of the model.

1. The world human population, \(P\) billions, is modelled by the equation

$$P = a b ^ { t }$$ where \(a\) and \(b\) are constants and \(t\) is the number of years after 2004
Using the estimated population figures for the years from 2004 to 2007, a graph is plotted of \(\log _ { 10 } P\) against \(t\). The points lie approximately on a straight line with

• gradient 0.0054
• intercept 0.81 on the \(\log _ { 10 } P\) axis
  1. Estimate, to 3 decimal places, the value of \(a\) and the value of \(b\).

In the context of the model,

1. interpret the value of the constant \(a\),
2. interpret the value of the constant \(b\).

Use the model to estimate the world human population in 2030

Comment on the reliability of the answer to part (c).

6 During some research the size, \(P\), of a population of insects, at time \(t\) months after the start of the research, is modelled by the following formula. \(P = 100 \mathrm { e } ^ { t }\)

1. Use this model to answer the following.
   1. Find the value of \(P\) when \(t = 4\).
   2. Find the value of \(t\) when the population is 9000 .
2. It is suspected that a more appropriate model would be the following formula. \(P = k a ^ { t }\) where \(k\) and \(a\) are constants.
   1. Show that, using this model, the graph of \(\log _ { 10 } P\) against \(t\) would be a straight line. Some observations of \(t\) and \(P\) gave the following results.

      \(t\)	1	2	3	4	5
      \(P\)	100	500	1800	7000	19000
      \(\log _ { 10 } P\)	2.00	2.70	3.26	3.85	4.28

   2. On the grid in the Printed Answer Booklet, draw a line of best fit for the data points \(\left( t , \log _ { 10 } P \right)\) given in the table.
   3. Hence estimate the values of \(k\) and \(a\).

9 A biologist is investigating the growth of bacteria in a piece of bread.
He believes that the number, \(N\), of bacteria after \(t\) hours may be modelled by the relationship \(N = A \times 2 ^ { k t }\), where \(A\) and \(k\) are constants.

1. Show that, according to the model, the graph of \(\log _ { 10 } N\) against \(t\) is a straight line. Give, in terms of \(A\) and \(k\),
   • the gradient of the line
   • the intercept on the vertical axis.
   The biologist measures the number of bacteria at regular intervals over 22 hours and plots a graph of \(\log _ { 10 } N\) against \(t\). He finds that the graph is approximately a straight line with gradient 0.20 . The line crosses the vertical axis at 2.0 .
2. Find the values of \(A\) and \(k\).
3. Use the model to predict the number of bacteria after 24 hours.
4. Give a reason why the model may not be appropriate for large values of \(t\).

14 Alex places a hot object into iced water and records the temperature \(\theta ^ { \circ } \mathrm { C }\) of the object every minute. The temperature of an object \(t\) minutes after being placed in iced water is modelled by \(\theta = \theta _ { 0 } \mathrm { e } ^ { - k t }\) where \(\theta _ { 0 }\) and \(k\) are constants whose values depend on the characteristics of the object. The temperature of Alex's object is \(82 ^ { \circ } \mathrm { C }\) when it is placed into the water. After 5 minutes the temperature is \(27 ^ { \circ } \mathrm { C }\).

1. Find the values of \(\theta _ { 0 }\) and \(k\) that best model the data.
2. Explain why the model may not be suitable in the long term if Alex does not top up the ice in the water.
3. Show that the model with the values found in part (a) can be written as \(\ln \theta = \mathrm { a } -\) bt where \(a\) and \(b\) are constants to be determined. Ben places a different object into iced water at the same time as Alex. The model for Ben's object is \(\ln \theta = 3.4 - 0.08 t\).
4. Determine each of the following:
   • the initial temperature of Ben's object
   • the rate at which Ben's object is cooling initially.
   • According to the models, there is a time at which both objects have the same temperature.
   Find this time and the corresponding temperature.

5 [Figure 1 and Figure 2, printed on the insert, are provided for use in this question.] The variables \(x\) and \(y\) are known to be related by an equation of the form $$y = a b ^ { x }$$ where \(a\) and \(b\) are constants. The following approximate values of \(x\) and \(y\) have been found.

1. Complete the table in Figure 1, showing values of \(x\) and \(Y\), where \(Y = \log _ { 10 } y\). Give each value of \(Y\) to three decimal places.
2. Show that, if \(y = a b ^ { x }\), then \(x\) and \(Y\) must satisfy an equation of the form $$Y = m x + c$$
3. Draw on Figure 2 a linear graph relating \(x\) and \(Y\).
4. Hence find estimates for the values of \(a\) and \(b\).

3 The variables \(y\) and \(x\) are related by an equation of the form $$y = a \left( b ^ { x } \right)$$ 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. The graph of \(Y\) against \(x\), shown below, passes through the points ( \(0,2.5\) ) and (5, 0.5). \includegraphics[max width=\textwidth, alt={}, center]{7e7eaea5-22ca-4418-8ac6-351ce9ac09ea-06_433_506_904_776}
   1. Find the gradient of the line.
   2. Find the value of \(a\) and the value of \(b\), giving each answer to three significant figures. [4 marks]

13 Answer part (ii) of this question on the insert provided. The table gives a firm's monthly profits for the first few months after the start of its business, rounded to the nearest \(\pounds 100\). The firm's profits, \(\pounds y\), for the \(x\) th month after start-up are modelled by $$y = k \times 10 ^ { a x }$$ where \(a\) and \(k\) are constants.

1. Show that, according to this model, a graph of \(\log _ { 10 } y\) against \(x\) gives a straight line of gradient \(a\) and intercept \(\log _ { 10 } k\).
2. On the insert, complete the table and plot \(\log _ { 10 } y\) against \(x\), drawing by eye a line of best fit.
3. Use your graph to find an equation for \(y\) in terms of \(x\) for this model.
4. For which month after start-up does this model predict profits of about \(\pounds 75000\) ?
5. State one way in which this model is unrealistic.

9 The table below shows the annual global production of plastics, \(P\), measured in millions of tonnes per year, for six selected years. It is thought that \(P\) can be modelled by $$P = A \times 10 ^ { k t }$$ where \(t\) is the number of years after 1980 and \(A\) and \(k\) are constants.
9

1. Show algebraically that the graph of \(\log _ { 10 } P\) against \(t\) should be linear.
   9
2. 1. Complete the table below.

      \(\boldsymbol { t }\)	0	5	10	15	20	25
      \(\boldsymbol { \operatorname { l o g } } _ { \mathbf { 1 0 } } \boldsymbol { P }\)	1.88	1.97	2.08		2.31

      9
3. (ii) Plot \(\log _ { 10 } P\) against \(t\), and draw a line of best fit for the data. \includegraphics[max width=\textwidth, alt={}, center]{042e248a-9efa-4844-957d-f05715900ffc-13_1203_1308_360_367} 9
4. 1. Hence, show that \(k\) is approximately 0.02
      9
5. (ii) Find the value of \(A\).
   9
6. Using the model with \(k = 0.02\) predict the number of tonnes of annual global production of plastics in 2030. 9
7. Using the model with \(k = 0.02\) predict the year in which \(P\) first exceeds 8000
   9
8. Give a reason why it may be inappropriate to use the model to make predictions about future annual global production of plastics. \includegraphics[max width=\textwidth, alt={}, center]{042e248a-9efa-4844-957d-f05715900ffc-15_2488_1716_219_153}

8 Theresa bought a house on 2 January 1970 for \(\pounds 8000\). 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. The valuation price of the house can be modelled by the equation $$V = p q ^ { t }$$ where \(V\) pounds is the valuation price \(t\) years after 2 January 1970 and \(p\) and \(q\) are constants. 8

1. Show that \(V = p q ^ { t }\) can be written as \(\log _ { 10 } V = \log _ { 10 } p + t \log _ { 10 } q\) 8
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.

   \(t\)	0	10	20	30	40
   \(\log _ { 10 } V\)	3.90	4.28	4.56	4.91	5.31

   \includegraphics[max width=\textwidth, alt={}]{838f0625-95e6-4ad4-b97b-3d3f77cc7f19-11_1927_1207_625_338}
   Using the given line of best fit, find estimates for the values of \(p\) and \(q\). Give your answers correct to three significant figures.
   8
3. Determine the year in which Theresa's house will first be worth half a million pounds. 8
4. Explain whether your answer to part (c) is likely to be reliable.