1.06i Exponential growth/decay: in modelling context

12. The table shows the average weekly pay of a footballer at a certain club on 1 August 1990 and 1 August 2010. The average weekly pay of a footballer at this club can be modelled by the equation $$P = A k ^ { t }$$ where \(\pounds P\) is the average weekly pay \(t\) years after 1 August 1990, and \(A\) and \(k\) are constants.
a. i. Write down the value of \(A\).
ii. Show that the value of \(k\) is 1.16159 , correct to five decimal places.
b. With reference to the model, interpret
i. the value of the constant \(A\),
ii. the value of the constant \(k\), Using the model,
c. find the year in which, on 1 August, the average weekly pay of a footballer at this club will first exceed \(\pounds 100000\).

9. A cup of tea is cooling down in a room. The temperature of tea, \(\theta ^ { \circ } \mathrm { C }\), at time \(t\) minutes after the tea is made, is modelled by the equation $$\theta = A + 70 e ^ { - 0.025 t }$$ where \(A\) is a positive constant.
Given that the initial temperature of the tea is \(85 ^ { \circ } \mathrm { C }\) a. find the value of \(A\).
b. Find the temperature of the tea 20 minutes after it is made.
c. Find how long it will take the tea to cool down to \(43 ^ { \circ } \mathrm { C }\).
(4)

14. A population of ants being studied on an island. The number of ants, \(P\), in the population, is modelled by the equation. $$P = \frac { 900 k e ^ { 0.2 t } } { 1 + k e ^ { 0.2 t } } , \text { where } k \text { is a constant. }$$ Given that there were 360 ants when the study started,
a. show that \(k = \frac { 2 } { 3 }\).
b. Show that \(P = \frac { 1800 } { 2 + 3 e ^ { - 0.2 t } }\). The model predicts an upper limit to the number of ants on the island.
c. State the value of this limit.
d. Find the value of \(t\) when \(P = 520\). Give your answer to one decimal place.
e. i. Show that the rate of growth, \(\frac { \mathrm { d } P } { d t } = \frac { P ( 900 - P ) } { 4500 }\) ii. Hence state the value of \(P\) at which the rate of growth is a maximum.

1. The value, \(\pounds V\), of a vintage car \(t\) years after it was first valued on 1 st January 2001, is modelled by the equation

$$V = A p ^ { t } \quad \text { where } A \text { and } p \text { are constants }$$ Given that the value of the car was \(\pounds 32000\) on 1st January 2005 and \(\pounds 50000\) on 1st January 2012

1. 1. find \(p\) to 4 decimal places,
   2. show that \(A\) is approximately 24800
2. With reference to the model, interpret
   1. the value of the constant \(A\),
   2. the value of the constant \(p\). Using the model,
3. find the year during which the value of the car first exceeds \(\pounds 100000\)

1. In a simple model, the value, \(\pounds V\), of a car depends on its age, \(t\), in years.

The following information is available for \(\operatorname { car } A\)

• its value when new is \(\pounds 20000\)
• its value after one year is \(\pounds 16000\)
  1. Use an exponential model to form, for car \(A\), a possible equation linking \(V\) with \(t\).

The value of car \(A\) is monitored over a 10-year period.
Its value after 10 years is \(\pounds 2000\)

Evaluate the reliability of your model in light of this information. The following information is available for car \(B\)

• it has the same value, when new, as car \(A\)
• its value depreciates more slowly than that of \(\operatorname { car } A\)
• Explain how you would adapt the equation found in (a) so that it could be used to model the value of car \(B\).

1. A scientist is studying the number of bees and the number of wasps on an island.

The number of bees, measured in thousands, \(N _ { b }\), is modelled by the equation $$N _ { b } = 45 + 220 \mathrm { e } ^ { 0.05 t }$$ where \(t\) is the number of years from the start of the study.
According to the model,

1. find the number of bees at the start of the study,
2. show that, exactly 10 years after the start of the study, the number of bees was increasing at a rate of approximately 18 thousand per year. The number of wasps, measured in thousands, \(N _ { w }\), is modelled by the equation $$N _ { w } = 10 + 800 \mathrm { e } ^ { - 0.05 t }$$ where \(t\) is the number of years from the start of the study.
   When \(t = T\), according to the models, there are an equal number of bees and wasps.
3. Find the value of \(T\) to 2 decimal places.

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.

7. \begin{figure}[h] \end{figure} Figure 2 Figure 2 shows a cylindrical tank of height 1.5 m .
Initially the tank is full of water.
The water starts to leak from a small hole, at a point \(L\), in the side of the tank.
While the tank is leaking, the depth, \(H\) metres, of the water in the tank is modelled by the differential equation $$\frac { \mathrm { d } H } { \mathrm {~d} t } = - 0.12 \mathrm { e } ^ { - 0.2 t }$$ where \(t\) hours is the time after the leak starts.
Using the model,

1. show that $$H = A \mathrm { e } ^ { - 0.2 t } + B$$ where \(A\) and \(B\) are constants to be found,
2. find the time taken for the depth of the water to decrease to 1.2 m . Give your answer in hours and minutes, to the nearest minute. In the long term, the water level in the tank falls to the same height as the hole.
3. Find, according to the model, the height of the hole from the bottom of the tank.

5. The mass, \(m\) grams, of a radioactive substance, \(t\) years after first being observed, is modelled by the equation $$m = 25 \mathrm { e } ^ { - 0.05 t }$$ According to the model,

1. find the mass of the radioactive substance six months after it was first observed,
2. show that \(\frac { \mathrm { d } m } { \mathrm {~d} t } = k m\), where \(k\) is a constant to be found.

1. A cup of hot tea was placed on a table. At time \(t\) minutes after the cup was placed on the table, the temperature of the tea in the cup, \(\theta ^ { \circ } \mathrm { C }\), is modelled by the equation

$$\theta = 25 + A \mathrm { e } ^ { - 0.03 t }$$ where \(A\) is a constant. The temperature of the tea was \(75 ^ { \circ } \mathrm { C }\) when the cup was placed on the table.

1. Find a complete equation for the model.
2. Use the model to find the time taken for the tea to cool from \(75 ^ { \circ } \mathrm { C }\) to \(60 ^ { \circ } \mathrm { C }\), giving your answer in minutes to one decimal place. Two hours after the cup was placed on the table, the temperature of the tea was measured as \(20.3 ^ { \circ } \mathrm { C }\). Using this information,
3. evaluate the model, explaining your reasoning.

1. A bacterial culture has area \(p \mathrm {~mm} ^ { 2 }\) at time \(t\) hours after the culture was placed onto a circular dish.

A scientist states that at time \(t\) hours, the rate of increase of the area of the culture can be modelled as being proportional to the area of the culture.

1. Show that the scientist's model for \(p\) leads to the equation $$p = a \mathrm { e } ^ { k t }$$ where \(a\) and \(k\) are constants. The scientist measures the values for \(p\) at regular intervals during the first 24 hours after the culture was placed onto the dish. She plots a graph of \(\ln p\) against \(t\) and finds that the points on the graph lie close to a straight line with gradient 0.14 and vertical intercept 3.95
2. Estimate, to 2 significant figures, the value of \(a\) and the value of \(k\).
3. Hence show that the model for \(p\) can be rewritten as $$p = a b ^ { t }$$ stating, to 3 significant figures, the value of the constant \(b\). With reference to this model,
4. 1. interpret the value of the constant \(a\),
   2. interpret the value of the constant \(b\).
5. State a long term limitation of the model for \(p\).

6 A pan of water is heated until it reaches \(100 ^ { \circ } \mathrm { C }\). Once the water reaches \(100 ^ { \circ } \mathrm { C }\), the heat is switched off and the temperature \(T ^ { \circ } \mathrm { C }\) of the water decreases. The temperature of the water is modelled by the equation $$T = 25 + a \mathrm { e } ^ { - k t }$$ where \(t\) denotes the time, in minutes, after the heat is switched off and \(a\) and \(k\) are positive constants.

1. Write down the value of \(a\).
2. Explain what the value of 25 represents in the equation \(T = 25 + a \mathrm { e } ^ { - k t }\). When the heat is switched off, the initial rate of decrease of the temperature of the water is \(15 ^ { \circ } \mathrm { C }\) per minute.
3. Calculate the value of \(k\).
4. Find the time taken for the temperature of the water to drop from \(100 ^ { \circ } \mathrm { C }\) to \(45 ^ { \circ } \mathrm { C }\).
5. A second pan of water is heated, but the heat is turned off when the water is at a temperature of less than \(100 ^ { \circ } \mathrm { C }\). Suggest how the equation for the temperature as the water cools would be modified by this.

10 Layla invests money in the bank and receives compound interest. The amount \(\pounds L\) that she has after \(t\) years is given by the equation \(\mathrm { L } = 2800 \times 1.023 ^ { \mathrm { t } }\).

1. 1. State the amount she invests.
   2. State the annual rate of interest. Amit invests \(\pounds 3000\) and receives \(2 \%\) compound interest per year. The amount \(\pounds A\) that he has after \(t\) years is given by the equation \(\mathrm { A } = \mathrm { ab } ^ { \mathrm { t } }\).
2. Determine the values of the constants \(a\) and \(b\).
3. Layla and Amit invest their money in the bank at the same time. Determine the value of \(t\) for which Layla and Amit have equal amounts in the bank. Give your answer correct to \(\mathbf { 1 }\) decimal place.

11 On the day that a new consumer product went on sale (day zero), a call centre received 1 call about it. On the 2nd day after day zero the call centre received 3 calls, and on the 10th day after day zero there were 200 calls. Two models were proposed to model \(N\), the number of calls received \(t\) days after day zero.
Model 1 is a linear model \(\mathrm { N } = \mathrm { mt } + \mathrm { c }\).

1. Determine the values of \(m\) and \(c\) which best model the data for 2 days and 10 days after day zero.
2. State the rate of increase in calls according to model 1.
3. Explain why this model is not suitable when \(t = 1\). Model 2 is an exponential model \(\mathbf { N } = e ^ { 0.53 t }\).
4. Verify that this is a good model for the number of calls when \(t = 2\) and \(t = 10\).
5. Determine the rate of increase in calls when \(t = 10\) according to model 2 .

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 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 \(t\).

11 A car is travelling along a stretch of road at a steady speed of \(11 \mathrm {~ms} ^ { - 1 }\).
The driver accelerates, and \(t\) seconds after starting to accelerate the speed of the car, \(V\), is modelled by the formula \(\mathrm { V } = \mathrm { A } + \mathrm { B } \left( 1 - \mathrm { e } ^ { - 0.17 \mathrm { t } } \right)\).
When \(t = 3 , V = 13.8\).

1. Find the values of \(A\) and \(B\), giving your answers correct to 2 significant figures. When \(t = 4 , V = 14.5\) and when \(t = 5 , V = 14.9\).
2. Determine whether the model is a good fit for these data.
3. Determine the acceleration of the car according to the model when \(t = 5\), giving your answer correct to 3 decimal places. The car continues to accelerate until it reaches its maximum speed.
   The speed limit on this road is \(60 \mathrm { kmh } ^ { - 1 }\). All drivers who exceed this speed limit are recorded by a speed camera and automatically fined \(\pounds 100\).
4. Determine whether, according to the model, the driver of this car is fined \(\pounds 100\).

15 A model for the motion of a small object falling through a thick fluid can be expressed using the differential equation \(\frac { \mathrm { d } v } { \mathrm {~d} t } = 9.8 - k v\),
where \(v \mathrm {~ms} ^ { - 1 }\) is the velocity after \(t \mathrm {~s}\) and \(k\) is a positive constant.

1. Given that \(v = 0\) when \(t = 0\), solve the differential equation to find \(v\) in terms of \(t\) and \(k\).
2. Sketch the graph of \(v\) against \(t\). Experiments show that for large values of \(t\), the velocity tends to \(7 \mathrm {~ms} ^ { - 1 }\).
3. Find the value of \(k\).
4. Find the value of \(t\) for which \(v = 3.5\).

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:
   Find this time and the corresponding temperature.

11 The height \(h \mathrm {~cm}\) of a sunflower plant \(t\) days after planting the seed is modelled by \(\mathrm { h } = \mathrm { a } + \mathrm { b }\) Int for \(t \geqslant 9\), where \(a\) and \(b\) are constants. The sunflower is 10 cm tall 10 days after planting and 200 cm tall 85 days after planting.

1. 1. Show that the value of \(b\) which best models these values is 88.8 correct to \(\mathbf { 3 }\) significant figures.
   2. Find the corresponding value of \(a\).
2. 1. Explain why the model is not suitable for small positive values of \(t\).
   2. Explain why the model is not suitable for very large positive values of \(t\).
3. Show that the model indicates that the sunflower grows to 1 m in height in less than half the time it takes to grow to 2 m .
4. Find the value of \(t\) for which the rate of growth is 3 cm per day.

10 Zac is measuring the growth of a culture of bacteria in a laboratory. The initial area of the culture is \(8 \mathrm {~cm} ^ { 2 }\). The area one day later is \(8.8 \mathrm {~cm} ^ { 2 }\). At first, Zac uses a model of the form \(\mathrm { A } = \mathrm { a } + \mathrm { bt }\), where \(A \mathrm {~cm} ^ { 2 }\) is the area \(t\) days after he begins measuring and \(a\) and \(b\) are constants.

1. Find the values of \(a\) and \(b\) that best model the initial area and the area one day later.
2. Calculate the value of \(t\) for which the model predicts an area of \(15 \mathrm {~cm} ^ { 2 }\).
3. Zac notices the area covered by the culture increases by \(10 \%\) each day. Explain why this model may not be suitable after the first day. Zac decides to use a different model for \(A\). His new model is \(\mathrm { A } = \mathrm { Pe } ^ { \mathrm { kt } }\), where \(P\) and \(k\) are constants.
4. Find the values of \(P\) and \(k\) that best model the initial area and the area one day later.
5. Calculate the value of \(t\) for which the area reaches \(15 \mathrm {~cm} ^ { 2 }\) according to this model.
6. Explain why this model may not be suitable for large values of \(t\).

10 In a certain region, the populations of grey squirrels, \(P _ { \mathrm { G } }\) and red squirrels \(P _ { \mathrm { R } }\), at time \(t\) years are modelled by the equations: \(P _ { \mathrm { G } } = 10000 \left( 1 - \mathrm { e } ^ { - k t } \right)\) \(P _ { \mathrm { R } } = 20000 \mathrm { e } ^ { - k t }\) where \(t \geq 0\) and \(k\) is a positive constant.

1. 1. On the axes in your Printed Answer Book, sketch the graphs of \(P _ { \mathrm { G } }\) and \(P _ { \mathrm { R } }\) on the same axes.
   2. Give the equations of any asymptotes.
2. What does the model predict about the long term population of
   Grey squirrels and red squirrels compete for food and space. Grey squirrels are larger and more successful than red squirrels.
3. Comment on the validity of the model given by the equations, giving a reason for your answer.
4. Show that, according to the model, the rate of decrease of the population of red squirrels is always double the rate of increase of the population of grey squirrels.
5. When \(t = 3\), the numbers of grey and red squirrels are equal. Find the value of \(k\).

9 A small country started using solar panels to produce electrical energy in the year 2000. Electricity production is measured in megawatt hours (MWh). For the period from 2000 to 2009, the annual electrical energy produced using solar panels can be modelled by the equation \(\mathrm { P } = 0.3 \mathrm { e } ^ { 0.5 \mathrm { t } }\), where \(P\) is the annual amount of electricity produced in MWh and \(t\) is the time in years after the year 2000.

1. According to this model, find the amount of electricity produced using solar panels in each of the following years.
   1. 2000
   2. 2009
2. Give a reason why the model is unlikely to be suitable for predicting the annual amount of electricity produced using solar panels in the year 2025. An alternative model is suggested; the curve representing this model is shown in Fig. 9. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Fig. 9} \includegraphics[alt={},max width=\textwidth]{20639e13-01cc-4d96-b694-fb3cf1828f4d-08_702_1587_1265_230}
   \end{figure}
3. Explain how the graph shows that the alternative model gives a value for the amount of electricity produced in 2009 that is consistent with the original model.
4. 1. On the axes given in the Printed Answer Booklet, sketch the gradient function of the model shown in Fig. 9.
   2. State approximately the value of \(t\) at the point of inflection in Fig. 9.
   3. Interpret the significance of the point of inflection in the context of the model.
5. State approximately the long term value of the annual amount of electricity produced using solar panels according to the model represented in Fig. 9.

6

1. 1. Write down the derivative of \(\mathrm { e } ^ { \mathrm { kx } }\), where \(k\) is a constant.
   2. A business has been running since 2009. They sell maths revision resources online. Give a reason why an exponential growth model might be suitable for the annual profits for the business. Fig. 6 shows the relationship between the annual profits of the business in thousands of pounds ( \(y\) ) and the time in years after \(2009 ( x )\). The graph of lny plotted against \(x\) is approximately a straight line. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{a13f7a05-e2d3-4354-a0c7-ef7283eff514-07_1052_1157_751_242} \captionsetup{labelformat=empty} \caption{Fig. 6}
      \end{figure}
2. Show that the straight line is consistent with a model of the form \(\mathbf { y } = \mathrm { Ae } ^ { \mathrm { kx } }\), where \(A\) and \(k\) are constants.
3. Estimate the values of \(A\) and \(k\).
4. Use the model to predict the profit in the year 2020.
5. How reliable do you expect the prediction in part (d) to be? Justify your answer.

5

1. The diagram shows the curve \(\mathrm { y } = \mathrm { e } ^ { \mathrm { x } }\). \includegraphics[max width=\textwidth, alt={}, center]{a0d9573f-8273-4562-a2d3-07f15d9da1af-5_574_682_315_328} On the axes in the Printed Answer Booklet, sketch graphs of
   1. \(\frac { \mathrm { dy } } { \mathrm { dx } }\) against \(x\),
   2. \(\frac { \mathrm { dy } } { \mathrm { dx } }\) against \(y\).
2. Wolves were introduced to Yellowstone National Park in 1995. The population of wolves, \(y\), is modelled by the equation \(y = A e ^ { k t }\),
   where \(A\) and \(k\) are constants and \(t\) is the number of years after 1995.
   1. Give a reason why this model might be suitable for the population of wolves.
   2. When \(t = 0 , y = 21\) and when \(t = 1 , y = 51\). Find values of \(A\) and \(k\) consistent with the data.
   3. Give a reason why the model will not be a good predictor of wolf populations many years after 1995.

1. During one day, a biological culure is allowed to grow under controlled conditions.

At 8 a.m. the culture is estimated to contain 20000 bacteria. A model of the growth of the culture assumes that \(t\) hours after 8 a.m., the number of bacteria present, \(N\), is given by $$N = 20000 \times ( 1.06 ) ^ { t } .$$ Using this model,

1. find the number of bacteria present at 11 a.m.,
2. find, to the nearest minute, the time when the initial number of bacteria will have doubled.