1.06i Exponential growth/decay: in modelling context

7. \begin{figure}[h] \end{figure} Figure 1 shows a graph of the temperature of a room, \(T ^ { \circ } \mathrm { C }\), at time \(t\) minutes.
The temperature is controlled by a thermostat such that when the temperature falls to \(12 ^ { \circ } \mathrm { C }\), a heater is turned on until the temperature reaches \(18 ^ { \circ } \mathrm { C }\). The room then cools until the temperature again falls to \(12 ^ { \circ } \mathrm { C }\). For \(t\) in the interval \(10 \leq t \leq 60\), \(T\) is given by $$T = 5 + A \mathrm { e } ^ { - k t } ,$$ where \(A\) and \(k\) are constants.
Given that \(T = 18\) when \(t = 10\) and that \(T = 12\) when \(t = 60\),

1. show that \(k = 0.0124\) to 3 significant figures and find the value of \(A\),
2. find the rate at which the temperature of the room is decreasing when \(t = 20\). The temperature again reaches \(18 ^ { \circ } \mathrm { C }\) when \(t = 70\) and the graph for \(70 \leq t \leq 120\) is a translation of the graph for \(10 \leq t \leq 60\).
3. Find the value of the constant \(B\) such that for \(70 \leq t \leq 120\) $$T = 5 + B \mathrm { e } ^ { - k t } .$$

1. The population in thousands, \(P\), of a town at time \(t\) years after \(1 ^ { \text {st } }\) January 1980 is modelled by the formula

$$P = 30 + 50 \mathrm { e } ^ { 0.002 t }$$ Use this model to estimate

1. the population of the town on \(1 { } ^ { \text {st } }\) January 2010,
2. the year in which the population first exceeds 84000 . The population in thousands, \(Q\), of another town is modelled by the formula $$Q = 26 + 50 \mathrm { e } ^ { 0.003 t }$$
3. Show that the value of \(t\) when \(P = Q\) is a solution of the equation $$t = 1000 \ln \left( 1 + 0.08 \mathrm { e } ^ { - 0.002 t } \right) .$$
4. Use the iteration formula $$t _ { n + 1 } = 1000 \ln \left( 1 + 0.08 \mathrm { e } ^ { - 0.002 t _ { n } } \right)$$ with \(t _ { 0 } = 50\) to find \(t _ { 1 } , t _ { 2 }\) and \(t _ { 3 }\) and hence, the year in which the populations of these two towns will be equal according to these models.

5. The number of bacteria present in a culture at time \(t\) hours is modelled by the continuous variable \(N\) and the relationship $$N = 2000 \mathrm { e } ^ { k t } ,$$ where \(k\) is a constant. Given that when \(t = 3 , N = 18000\), find

1. the value of \(k\) to 3 significant figures,
2. how long it takes for the number of bacteria present to double, giving your answer to the nearest minute,
3. the rate at which the number of bacteria is increasing when \(t = 3\).

5 A model for the radioactive decay of a form of iodine is given by $$m = m _ { 0 } 2 ^ { - \frac { 1 } { 8 } t }$$ The mass of the iodine after \(t\) days is \(m\) grams. Its initial mass is \(m _ { 0 }\) grams.

1. Use the given model to find the mass that remains after 10 grams of this form of iodine have decayed for 14 days, giving your answer to the nearest gram.
2. A mass of \(m _ { 0 }\) grams of this form of iodine decays to \(\frac { m _ { 0 } } { 16 }\) grams in \(d\) days. Find the value of \(d\).
3. After \(n\) days, a mass of this form of iodine has decayed to less than \(1 \%\) of its initial mass. Find the minimum integer value of \(n\).

4 A scientist is testing models for the growth and decay of colonies of bacteria. For a particular colony, which is growing, the model is \(P = A \mathrm { e } ^ { \frac { 1 } { 8 } t }\), where \(P\) is the number of bacteria after a time \(t\) minutes and \(A\) is a constant.

1. This growing colony consists initially of 500 bacteria. Calculate the number of bacteria, according to the model, after one hour. Give your answer to the nearest thousand.
2. For a second colony, which is decaying, the model is \(Q = 500000 \mathrm { e } ^ { - \frac { 1 } { 8 } t }\), where \(Q\) is the number of bacteria after a time \(t\) minutes. Initially, the growing colony has 500 bacteria and, at the same time, the decaying colony has 500000 bacteria.
   1. Find the time at which the populations of the two colonies will be equal, giving your answer to the nearest 0.1 of a minute.
   2. The population of the growing colony will exceed that of the decaying colony by 45000 bacteria at time \(T\) minutes. Show that $$\left( \mathrm { e } ^ { \frac { 1 } { 8 } T } \right) ^ { 2 } - 90 \mathrm { e } ^ { \frac { 1 } { 8 } T } - 1000 = 0$$ and hence find the value of \(T\), giving your answer to one decimal place.
      (4 marks)

9 An analyst believes that the sales of a particular electronic device are growing exponentially. In 2015 the sales were 3.1 million devices and the rate of increase in the annual sales is 0.8 million devices per year.

1. Find a model to represent the annual sales, defining any variables used.
2. In 2017 the sales were 5.2 million devices. Determine whether this is consistent with the model in part (i).
3. The analyst uses the model in part (i) to predict the sales for 2025. Comment on the reliability of this prediction.

7 As a spherical snowball melts its volume decreases. The rate of decrease of the volume of the snowball at any given time is modelled as being proportional to its volume at that time. Initially the volume of the snowball is \(500 \mathrm {~cm} ^ { 3 }\) and the rate of decrease of its volume is \(20 \mathrm {~cm} ^ { 3 }\) per hour.

1. Find the time that this model would predict for the snowball's volume to decrease to \(250 \mathrm {~cm} ^ { 3 }\).
2. Write down one assumption made when using this model.
3. Comment on how realistic this model would be in the long term.

7. A particular species of orchid is being studied. The population \(p\) at time \(t\) years after the study started is assumed to be $$p = \frac { 2800 a \mathrm { e } ^ { 0.2 t } } { 1 + a \mathrm { e } ^ { 0.2 t } } , \text { where } a \text { is a constant. }$$ Given that there were 300 orchids when the study started,

1. show that \(a = 0.12\),
2. use the equation with \(a = 0.12\) to predict the number of years before the population of orchids reaches 1850 .
3. Show that \(p = \frac { 336 } { 0.12 + \mathrm { e } ^ { - 0.2 t } }\).
4. Hence show that the population cannot exceed 2800.

4 On 1 January 1900, a sculpture was valued at \(\pounds 80\).
When the sculpture was sold on 1 January 1956, its value was \(\pounds 5000\).
The value, \(\pounds V\), of the sculpture is modelled by the formula \(V = A k ^ { t }\), where \(t\) is the time in years since 1 January 1900 and \(A\) and \(k\) are constants.

1. Write down the value of \(A\).
2. Show that \(k \approx 1.07664\).
3. Use this model to:
   1. show that the value of the sculpture on 1 January 2006 will be greater than £200 000;
   2. find the year in which the value of the sculpture will first exceed \(\pounds 800000\).

4 David is researching changes in the selling price of houses. One particular house was sold on 1 January 1885 for \(\pounds 20\). Sixty years later, on 1 January 1945, it was sold for \(\pounds 2000\). David proposes a model $$P = A k ^ { t }$$ for the selling price, \(\pounds P\), of this house, where \(t\) is the time in years after 1 January 1885 and \(A\) and \(k\) are constants.

1. 1. Write down the value of \(A\).
   2. Show that, to six decimal places, \(k = 1.079775\).
   3. Use the model, with this value of \(k\), to estimate the selling price of this house on 1 January 2008. Give your answer to the nearest \(\pounds 1000\).
2. For another house, which was sold for \(\pounds 15\) on 1 January 1885, David proposes the model $$Q = 15 \times 1.082709 ^ { t }$$ for the selling price, \(\pounds Q\), of this house \(t\) years after 1 January 1885. Calculate the year in which, according to these models, these two houses would have had the same selling price.

9 A botanist is investigating the rate of growth of a certain species of toadstool. She observes that a particular toadstool of this type has a height of 57 millimetres at a time 12 hours after it begins to grow. She proposes the model \(h = A \left( 1 - \mathrm { e } ^ { - \frac { 1 } { 4 } t } \right)\), where \(A\) is a constant, for the height \(h\) millimetres of the toadstool, \(t\) hours after it begins to grow.

1. Use this model to:
   1. find the height of the toadstool when \(t = 0\);
   2. show that \(A = 60\), correct to two significant figures.
2. Use the model \(h = 60 \left( 1 - \mathrm { e } ^ { - \frac { 1 } { 4 } t } \right)\) to:
   1. show that the time \(T\) hours for the toadstool to grow to a height of 48 millimetres is given by $$T = a \ln b$$ where \(a\) and \(b\) are integers;
   2. show that \(\frac { \mathrm { d } h } { \mathrm {~d} t } = 15 - \frac { h } { 4 }\);
   3. find the height of the toadstool when it is growing at a rate of 13 millimetres per hour.
      (1 mark)

4 A car depreciates in value according to the model $$V = A k ^ { t }$$ where \(\pounds V\) is the value of the car \(t\) months from when it was new, and \(A\) and \(k\) are constants. Its value when new was \(\pounds 12499\) and 36 months later its value was \(\pounds 7000\).

1. 1. Write down the value of \(A\).
   2. Show that the value of \(k\) is 0.984025 , correct to six decimal places.
2. The value of this car first dropped below \(\pounds 5000\) during the \(n\)th month from new. Find the value of \(n\).

6 A number of cases of the general exponential model for the marathon are given in Table 6. One of these is $$R = 115 + ( 175 - 115 ) \mathrm { e } ^ { - 0.0467 t ^ { 0.797 } }$$

1. What is the value of \(t\) for the year 2012?
2. What record time does this model predict for the year 2012?
3. \(\_\_\_\_\)
4. \(\_\_\_\_\)

10 A bottle of water has a temperature of \(6 ^ { \circ } \mathrm { C }\) when it is removed from a refrigerator. It is placed in a room where the temperature is \(20 ^ { \circ } \mathrm { C }\) 10 minutes later, the temperature of the water is \(12 ^ { \circ } \mathrm { C }\) The temperature of the water, \(T ^ { \circ } \mathrm { C }\), at time \(t\) minutes after it is removed from the refrigerator, may be modelled by the equation $$T = 20 - a \mathrm { e } ^ { - k t }$$ 10

1. Find the value of \(a\). 10
2. Calculate the value of \(k\), giving your answer to two significant figures.
   10
3. Using this model, estimate how long it takes the water to reach a temperature of \(18 ^ { \circ } \mathrm { C }\) after it is taken out of the refrigerator. \(18 ^ { \circ } \mathrm { C }\) after it is taken out of the refrigerator. 10
4. Explain why the model may not be appropriate to predict the temperature of the water three hours after it is taken out of the refrigerator.

10 A scientist is researching the effects of caffeine. She models the mass of caffeine in the body using $$m = m _ { 0 } \mathrm { e } ^ { - k t }$$ where \(m _ { 0 }\) milligrams is the initial mass of caffeine in the body and \(m\) milligrams is the mass of caffeine in the body after \(t\) hours. On average, it takes 5.7 hours for the mass of caffeine in the body to halve.
One cup of strong coffee contains 200 mg of caffeine.
10

1. The scientist drinks two strong cups of coffee at 8 am. Use the model to estimate the mass of caffeine in the scientist's body at midday.
   10
2. The scientist wants the mass of caffeine in her body to stay below 480 mg

   10 (b)

   Use the model to find the earliest time

   coffee.

   Give your answer to the nearest minute

   □

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. (i) 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 (b) (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
3. (i) Hence, show that \(k\) is approximately 0.02
   9 (c) (ii) Find the value of \(A\).
   9
4. Using the model with \(k = 0.02\) predict the number of tonnes of annual global production of plastics in 2030. 9
5. Using the model with \(k = 0.02\) predict the year in which \(P\) first exceeds 8000
   9
6. 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}

10 A gardener has a greenhouse containing 900 tomato plants. The gardener notices that some of the tomato plants are damaged by insects.
Initially there are 25 damaged tomato plants.
The number of tomato plants damaged by insects is increasing by \(32 \%\) each day.
10

1. The total number of plants damaged by insects, \(x\), is modelled by $$x = A \times B ^ { t }$$ where \(A\) and \(B\) are constants and \(t\) is the number of days after the gardener first noticed the damaged plants. 10
   1. (i) Use this model to find the total number of plants damaged by insects 5 days after the gardener noticed the damaged plants.
      10
   2. (ii) Explain why this model is not realistic in the long term.
      10
   3. A refined model assumes the rate of increase of the number of plants damaged by insects is given by $$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { x ( 900 - x ) } { 2700 }$$ 10
   4. 1. Show that $$\int \left( \frac { A } { x } + \frac { B } { 900 - x } \right) \mathrm { d } x = \int \mathrm { d } t$$ where \(A\) and \(B\) are positive integers to be found.
         

         10 (iii) Hence, find the number of days it takes from when the damage is first noticed until half of the plants are damaged by the insects. [2 marks] \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)

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
   1. (i) Use the equation of Victoria's line of best fit to show that, correct to three significant figures, \(a = 57.5\) [0pt] [1 mark]
      6
   2. (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
   3. According to Victoria's model, state the yearly percentage increase in the average weekly value of online grocery sales. 6
   4. 1. Use Victoria's model to predict the average weekly value of online grocery sales in 2025.
         6
   5. (ii) Explain why the prediction made in part (c)(i) may be unreliable.

1. The number of bacteria on a surface is being monitored.

The number of bacteria, \(N\), on the surface, \(t\) hours after monitoring began is modelled by the equation $$\log _ { 10 } N = 0.35 t + 2$$ Use the equation of the model to answer parts (a) to (c).

1. Find the initial number of bacteria on the surface.
2. Show that the equation of the model can be written in the form $$N = a b ^ { t }$$ where \(a\) and \(b\) are constants to be found. Give the value of \(b\) to 2 decimal places.
3. Hence find the rate of growth of bacteria on the surface exactly 5 hours after monitoring began.

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.

6 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 \(3 \mathrm { e } ^ { - 0.02 t }\) units and the concentration of Coldcure is \(5 \mathrm { e } ^ { - 0.07 t }\) units. Each drug becomes ineffective when its concentration falls below 1 unit.

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

A scientist is using carbon-14 dating to determine the age of some wooden items. The equation for carbon-14 dating an item is given by $$N = k\lambda^t$$ where

• \(N\) grams is the amount of carbon-14 currently present in the item
• \(k\) grams was the initial amount of carbon-14 present in the item
• \(t\) is the number of years since the item was made
• \(\lambda\) is a constant, with \(0 < \lambda < 1\)

1. Sketch the graph of \(N\) against \(t\) for \(k = 1\) [2]

Given that it takes 5700 years for the amount of carbon-14 to reduce to half its initial value,

1. show that the value of the constant \(\lambda\) is 0.999878 to 6 decimal places. [2]

Given that Item A

• is known to have had 15 grams of carbon-14 present initially
• is thought to be 3250 years old

1. calculate, to 3 significant figures, how much carbon-14 the equation predicts is currently in Item A. [2]

Item B is known to have initially had 25 grams of carbon-14 present, but only 18 grams now remain.

1. Use algebra to calculate the age of Item B to the nearest 100 years. [3]

The current, \(x\) amps, at time \(t\) seconds after a switch is closed in a particular electric circuit is modelled by the equation $$\frac{dx}{dt} = k - 3x$$ where \(k\) is a constant. Initially there is zero current in the circuit.

1. Solve the differential equation to find an equation, in terms of \(k\), for the current in the circuit at time \(t\) seconds. Give your answer in the form \(x = f(t)\). [6]

Given that in the long term the current in the circuit approaches \(7\) amps,

1. find the value of \(k\). [2]
2. Hence find the time in seconds it takes for the current to reach \(5\) amps, giving your answer to \(2\) significant figures. [3]

Every £1 of money invested in a savings scheme continuously gains interest at a rate of 4% per year. Hence, after \(x\) years, the total value of an initial £1 investment is £\(y\), where $$y = 1.04^x.$$

1. Sketch the graph of \(y = 1.04^x\), \(x \geq 0\). [2]
2. Calculate, to the nearest £, the total value of an initial £800 investment after 10 years. [2]
3. Use logarithms to find the number of years it takes to double the total value of any initial investment. [3]

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]