1.06i Exponential growth/decay: in modelling context

9 The temperature of a quantity of liquid at time \(t\) is \(\theta\). The liquid is cooling in an atmosphere whose temperature is constant and equal to \(A\). The rate of decrease of \(\theta\) is proportional to the temperature difference \(( \theta - A )\). Thus \(\theta\) and \(t\) satisfy the differential equation $$\frac { \mathrm { d } \theta } { \mathrm {~d} t } = - k ( \theta - A )$$ where \(k\) is a positive constant.

1. Find, in any form, the solution of this differential equation, given that \(\theta = 4 A\) when \(t = 0\).
2. Given also that \(\theta = 3 A\) when \(t = 1\), show that \(k = \ln \frac { 3 } { 2 }\).
3. Find \(\theta\) in terms of \(A\) when \(t = 2\), expressing your answer in its simplest form.

4

1. By sketching a suitable pair of graphs, show that the equation $$\ln x = 4 - \frac { 1 } { 2 } x$$ has exactly one real root, \(\alpha\).
2. Verify by calculation that \(4.5 < \alpha < 5.0\).
3. Use the iterative formula \(x _ { n + 1 } = 8 - 2 \ln x _ { n }\) to find \(\alpha\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

3 \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-04_527_634_255_717} The number of bacteria in a population, \(P\), at time \(t\) hours is modelled by the equation \(P = a \mathrm { e } ^ { k t }\), where \(a\) and \(k\) are constants. The graph of \(\ln P\) against \(t\), shown in the diagram, has gradient \(\frac { 1 } { 20 }\) and intersects the vertical axis at \(( 0,3 )\).

1. State the value of \(k\) and find the value of \(a\) correct to 2 significant figures.
2. Find the time taken for \(P\) to double. Give your answer correct to the nearest hour. \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-05_2723_33_99_22}

1. A population of a rare species of toad is being studied.

The number of toads, \(N\), in the population, \(t\) years after the start of the study, is modelled by the equation $$N = \frac { 900 \mathrm { e } ^ { 0.12 t } } { 2 \mathrm { e } ^ { 0.12 t } + 1 } \quad t \geqslant 0 , t \in \mathbb { R }$$ According to this model,

1. calculate the number of toads in the population at the start of the study,
2. find the value of \(t\) when there are 420 toads in the population, giving your answer to 2 decimal places.
3. Explain why, according to this model, the number of toads in the population can never reach 500

5. The temperature, \(\theta ^ { \circ } \mathrm { C }\), inside an oven, \(t\) minutes after the oven is switched on, is given by $$\theta = A - 180 \mathrm { e } ^ { - k t }$$ where \(A\) and \(k\) are positive constants. Given that the temperature inside the oven is initially \(18 ^ { \circ } \mathrm { C }\),

1. find the value of \(A\). The temperature inside the oven, 5 minutes after the oven is switched on, is \(90 ^ { \circ } \mathrm { C }\).
2. Show that \(k = p \ln q\) where \(p\) and \(q\) are rational numbers to be found. Hence find
3. the temperature inside the oven 9 minutes after the oven is switched on, giving your answer to 3 significant figures,
4. the rate of increase of the temperature inside the oven 9 minutes after the oven is switched on. Give your answer in \({ } ^ { \circ } \mathrm { C } \min ^ { - 1 }\) to 3 significant figures.

1. The percentage, \(P\), of the population of a small country who have access to the internet, is modelled by the equation

$$P = a b ^ { t }$$ where \(a\) and \(b\) are constants and \(t\) is the number of years after the start of 2005
Using the data for the years between the start of 2005 and the start of 2010, a graph is plotted of \(\log _ { 10 } P\) against \(t\). The points are found to lie approximately on a straight line with gradient 0.09 and intercept 0.68 on the \(\log _ { 10 } P\) axis.

1. Find, according to the model, the value of \(a\) and the value of \(b\), giving your answers to 2 decimal places.
2. In the context of the model, give a practical interpretation of the constant \(a\).
3. Use the model to estimate the percentage of the population who had access to the internet at the start of 2015

4. The growth of a weed on the surface of a pond is being studied. The surface area of the pond covered by the weed, \(A \mathrm {~m} ^ { 2 }\), is modelled by the equation $$A = \frac { 80 p \mathrm { e } ^ { 0.15 t } } { p \mathrm { e } ^ { 0.15 t } + 4 }$$ where \(p\) is a positive constant and \(t\) is the number of days after the start of the study.
Given that

• \(30 \mathrm {~m} ^ { 2 }\) of the surface of the pond was covered by the weed at the start of the study
• \(50 \mathrm {~m} ^ { 2 }\) of the surface of the pond was covered by the weed \(T\) days after the start of the study
  1. show that \(p = 2.4\)
  2. find the value of \(T\), giving your answer to one decimal place.
     (Solutions relying entirely on graphical or numerical methods are not acceptable.)

The weed grows until it covers the surface of the pond.

Find, according to the model, the maximum possible surface area of the pond.

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 \(t = 5\)

1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

A population of fruit flies is being studied.
The number of fruit flies, \(F\), in the population, \(t\) days after the start of the study, is modelled by the equation $$F = \frac { 350 \mathrm { e } ^ { k t } } { 9 + \mathrm { e } ^ { k t } }$$ where \(k\) is a constant.
Use the equation of the model to answer parts (a), (b) and (c).

1. Find the number of fruit flies in the population at the start of the study. Given that there are 200 fruit flies in the population 15 days after the start of the study,
2. show that \(k = \frac { 1 } { 15 } \ln 12\) Given also that, when \(t = T\), the number of fruit flies in the population is increasing at a rate of 10 per day,
3. find the possible values of \(T\), giving your answers to one decimal place.

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.

1. In this question you must show all stages of your working.

\section*{Solutions relying entirely on calculator technology are not acceptable.} The temperature, \(T ^ { \circ } \mathrm { C }\), of the air in a room \(t\) minutes after a heat source is switched off, is modelled by the equation $$T = 10 + A \mathrm { e } ^ { - B t }$$ where \(A\) and \(B\) are constants.
Given that the temperature of the air in the room at the instant the heat source was switched off was \(18 ^ { \circ } \mathrm { C }\),

1. find the value of \(A\) Given also that, exactly 45 minutes after the heat source was switched off, the temperature of the air in the room was \(16 ^ { \circ } \mathrm { C }\),
2. find the value of \(B\) to 3 significant figures. Using the values for \(A\) and \(B\),
3. find, according to the model, the rate of change of the temperature of the air in the room exactly two minutes after the heat source was switched off.
   Give your answer in \({ } ^ { \circ } \mathrm { C } \min ^ { - 1 }\) to 3 significant figures.
4. Explain why, according to the model, the temperature of the air in the room cannot fall to \(5 ^ { \circ } \mathrm { C }\)

5. \begin{figure}[h] \end{figure} The growth of duckweed on a pond is being studied. The surface area of the pond covered by duckweed, \(A \mathrm {~m} ^ { 2 }\), at a time \(t\) days after the start of the study is modelled by the equation $$A = p q ^ { t } \quad \text { where } p \text { and } q \text { are positive constants }$$ Figure 1 shows the linear relationship between \(\log _ { 10 } A\) and \(t\).
The points \(( 0,0.32 )\) and \(( 8,0.56 )\) lie on the line as shown.

1. Find, to 3 decimal places, the value of \(p\) and the value of \(q\). Using the model with the values of \(p\) and \(q\) found in part (a),
2. find the rate of increase of the surface area of the pond covered by duckweed, in \(\mathrm { m } ^ { 2 }\) / day, exactly 6 days after the start of the study.
   Give your answer to 2 decimal places. \includegraphics[max width=\textwidth, alt={}, center]{76205772-5395-4ab2-96f9-ad9803b8388c-19_2649_1840_117_114}

8. A scientist is studying a population of fish in a lake. The number of fish, \(N\), in the population, \(t\) years after the start of the study, is modelled by the equation $$N = \frac { 600 \mathrm { e } ^ { 0.3 t } } { 2 + \mathrm { e } ^ { 0.3 t } } \quad t \geqslant 0$$ Use the equation of the model to answer parts (a), (b), (c), (d) and (e).

1. Find the number of fish in the lake at the start of the study.
2. Find the upper limit to the number of fish in the lake.
3. Find the time, after the start of the study, when there are predicted to be 500 fish in the lake. Give your answer in years and months to the nearest month.
4. Show that $$\frac { \mathrm { d } N } { \mathrm {~d} t } = \frac { A \mathrm { e } ^ { 0.3 t } } { \left( 2 + \mathrm { e } ^ { 0.3 t } \right) ^ { 2 } }$$ where \(A\) is a constant to be found. Given that when \(t = T , \frac { \mathrm {~d} N } { \mathrm {~d} t } = 8\)
5. find the value of \(T\) to one decimal place.
   (Solutions relying entirely on calculator technology are not acceptable.) \includegraphics[max width=\textwidth, alt={}, center]{76205772-5395-4ab2-96f9-ad9803b8388c-27_2644_1840_118_111}

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 is studying two different populations of bacteria.

The number of bacteria \(N\) in the first population is modelled by the equation $$N = A \mathrm { e } ^ { k t } \quad t \geqslant 0$$ where \(A\) and \(k\) are positive constants and \(t\) is the time in hours from the start of the study. Given that

• there were 2500 bacteria in this population at the start of the study
• there were 10000 bacteria 8 hours later
  1. find the exact value of \(A\) and the value of \(k\) to 4 significant figures.

The number of bacteria \(N\) in the second population is modelled by the equation $$N = 60000 \mathrm { e } ^ { - 0.6 t } \quad t \geqslant 0$$ where \(t\) is the time in hours from the start of the study.

Find the rate of decrease of bacteria in this population exactly 5 hours from the start of the study. Give your answer to 3 significant figures. When \(t = T\), the number of bacteria in the two different populations was the same.

Find the value of \(T\), giving your answer to 3 significant figures.
(Solutions relying entirely on calculator technology are not acceptable.)

3. \begin{figure}[h] \end{figure} The total mass of gold, \(G\) tonnes, extracted from a mine is modelled by the equation $$G = 40 - 30 \mathrm { e } ^ { 1 - 0.05 t } \quad t \geqslant k \quad G \geqslant 0$$ where \(t\) is the number of years after 1st January 1800.
Figure 2 shows a sketch of \(G\) against \(t\). Use the equation of the model to answer parts (a), (b) and (c).

1. 1. Find the value of \(k\).
   2. Hence find the year and month in which gold started being extracted from the mine.
2. Find the total mass of gold extracted from the mine up to 1st January 1870. There is a limit to the mass of gold that can be extracted from the mine.
3. State the value of this limit.
   M

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. A new mobile phone is released for sale.

The total sales \(N\) of this phone, in thousands, is modelled by the equation $$N = 125 - A \mathrm { e } ^ { - 0.109 t } \quad t \geqslant 0$$ where \(A\) is a constant and \(t\) is the time in months after the phone was released for sale.
Given that when \(t = 0 , N = 32\)

1. state the value of \(A\). Given that when \(t = T\) the total sales of the phone was 100000
2. find, according to the model, the value of \(T\). Give your answer to 2 decimal places.
3. Find, according to the model, the rate of increase in total sales when \(t = 7\), giving your answer to 3 significant figures.
   (Solutions relying entirely on calculator technology are not acceptable.) The total sales of the mobile phone is expected to reach 150000
   Using this information,
4. give a reason why the given equation is not suitable for modelling the total sales of the phone.

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 \(q\)

13. A scientist is studying a population of insects. The number of insects, \(N\), in the population, \(t\) days after the start of the study is modelled by the equation $$N = \frac { 240 } { 1 + k \mathrm { e } ^ { - \frac { t } { 16 } } }$$ where \(k\) is a constant.
Given that there were 50 insects at the start of the study,

1. find the value of \(k\)
2. use the model to find the value of \(t\) when \(N = 100\)
3. Show that $$\frac { \mathrm { d } N } { \mathrm {~d} t } = \frac { 1 } { p } N - \frac { 1 } { q } N ^ { 2 }$$ where \(p\) and \(q\) are integers to be found.
   

   END

5. A bath is filled with hot water. The temperature, \(\theta ^ { \circ } \mathrm { C }\), of the water in the bath, \(t\) minutes after the bath has been filled, is given by $$\theta = 20 + A \mathrm { e } ^ { - k t }$$ where \(A\) and \(k\) are positive constants. Given that the temperature of the water in the bath is initially \(38 ^ { \circ } \mathrm { C }\),

1. find the value of \(A\). The temperature of the water in the bath 16 minutes after the bath has been filled is \(24.5 ^ { \circ } \mathrm { C }\).
2. Show that \(k = \frac { 1 } { 8 } \ln 2\) Using the values for \(k\) and \(A\),
3. find the temperature of the water 40 minutes after the bath has been filled, giving your answer to 3 significant figures.
4. Explain why the temperature of the water in the bath cannot fall to \(19 ^ { \circ } \mathrm { C }\).
   

3. The number of bacteria in a liquid culture is modelled by the formula $$N = 3500 ( 1.035 ) ^ { t } , \quad t \geqslant 0$$ where \(N\) is the number of bacteria \(t\) hours after the start of a scientific study.

1. State the number of bacteria at the start of the scientific study.
   (1)
2. Find the time taken from the start of the study for the number of bacteria to reach 10000
   Give your answer in hours and minutes, to the nearest minute.
3. Use calculus to find the rate of increase in the number of bacteria when \(t = 8\) Give your answer, in bacteria per hour, to the nearest whole number.

5. The radioactive decay of a substance is given by $$R = 1000 \mathrm { e } ^ { - c t } , \quad t \geqslant 0 .$$ where \(R\) is the number of atoms at time \(t\) years and \(c\) is a positive constant.

1. Find the number of atoms when the substance started to decay. It takes 5730 years for half of the substance to decay.
2. Find the value of \(c\) to 3 significant figures.
3. Calculate the number of atoms that will be left when \(t = 22920\).
4. In the space provided on page 13, sketch the graph of \(R\) against \(t\).

4. Joan brings a cup of hot tea into a room and places the cup on a table. At time \(t\) minutes after Joan places the cup on the table, the temperature, \(\theta ^ { \circ } \mathrm { C }\), of the tea is modelled by the equation $$\theta = 20 + A \mathrm { e } ^ { - k t } ,$$ where \(A\) and \(k\) are positive constants. Given that the initial temperature of the tea was \(90 ^ { \circ } \mathrm { C }\),

1. find the value of \(A\). The tea takes 5 minutes to decrease in temperature from \(90 ^ { \circ } \mathrm { C }\) to \(55 ^ { \circ } \mathrm { C }\).
2. Show that \(k = \frac { 1 } { 5 } \ln 2\).
3. Find the rate at which the temperature of the tea is decreasing at the instant when \(t = 10\). Give your answer, in \({ } ^ { \circ } \mathrm { C }\) per minute, to 3 decimal places.

3. The area, \(A \mathrm {~mm} ^ { 2 }\), of a bacterial culture growing in milk, \(t\) hours after midday, is given by $$A = 20 \mathrm { e } ^ { 1.5 t } , \quad t \geqslant 0$$

1. Write down the area of the culture at midday.
2. Find the time at which the area of the culture is twice its area at midday. Give your answer to the nearest minute.