1.06i Exponential growth/decay: in modelling context

4 A cup of water is cooling. Its initial temperature is \(100 ^ { \circ } \mathrm { C }\). After 3 minutes, its temperature is \(80 ^ { \circ } \mathrm { C }\).

1. Given that \(T = 25 + a \mathrm { e } ^ { - k t }\), where \(T\) is the temperature in \({ } ^ { \circ } \mathrm { C } , t\) is the time in minutes and \(a\) and \(k\) are constants, find the values of \(a\) and \(k\).
2. What is the temperature of the water
   (A) after 5 minutes,
   (B) in the long term?

9. \begin{figure}[h] \end{figure} The population of a species of animal is being studied. The population \(P\), at time \(t\) years from the start of the study, is assumed to be $$P = \frac { 9000 \mathrm { e } ^ { k t } } { 3 \mathrm { e } ^ { k t } + 7 } , \quad t \geqslant 0$$ where \(k\) is a positive constant.
A sketch of the graph of \(P\) against \(t\) is shown in Figure 2 .
Use the given equation to

1. find the population at the start of the study,
2. find the value for the upper limit of the population. Given that \(P = 2500\) when \(t = 4\)
3. calculate the value of \(k\), giving your answer to 3 decimal places. Using this value for \(k\),
4. find, using \(\frac { \mathrm { d } P } { \mathrm {~d} t }\), the rate at which the population is increasing when \(t = 10\) Give your answer to the nearest integer.

12 The table shows the size of a population of house sparrows from 1980 to 2005. The 'red alert' category for birds is used when a population has decreased by at least \(50 \%\) in the previous 25 years.

1. Show that the information for this population is consistent with the house sparrow being on red alert in 2005. The size of the population may be modelled by a function of the form \(P = a \times 10 ^ { - k t }\), where \(P\) is the population, \(t\) is the number of years after 1980, and \(a\) and \(k\) are constants.
2. Write the equation \(P = a \times 10 ^ { - k t }\) in logarithmic form using base 10, giving your answer as simply as possible.
3. Complete the table and draw the graph of \(\log _ { 10 } P\) against \(t\), drawing a line of best fit by eye.
4. Use your graph to find the values of \(a\) and \(k\) and hence the equation for \(P\) in terms of \(t\).
5. Find the size of the population in 2015 as predicted by this model. Would the house sparrow still be on red alert? Give a reason for your answer.

5 The mass, \(M\) grams, of a certain substance is increasing exponentially so that, at time \(t\) hours, the mass is given by $$M = 40 \mathrm { e } ^ { k t }$$ where \(k\) is a constant. The following table shows certain values of \(t\) and \(M\).

1. In either order,
   1. find the values missing from the table,
   2. determine the value of \(k\).
   3. Find the rate at which the mass is increasing when \(t = 21\).

7

1. Substance \(A\) is decaying exponentially and its mass is recorded at regular intervals. At time \(t\) years, the mass, \(M\) grams, of substance \(A\) is given by $$M = 40 \mathrm { e } ^ { - 0.132 t }$$
   1. Find the time taken for the mass of substance \(A\) to decrease to \(25 \%\) of its value when \(t = 0\).
   2. Find the rate at which the mass of substance \(A\) is decreasing when \(t = 5\).
   3. Substance \(B\) is also decaying exponentially. Initially its mass was 40 grams and, two years later, its mass is 31.4 grams. Find the mass of substance \(B\) after a further year.

8 An experiment involves two substances, Substance 1 and Substance 2, whose masses are changing. The mass, \(M _ { 1 }\) grams, of Substance 1 at time \(t\) hours is given by $$M _ { 1 } = 400 \mathrm { e } ^ { - 0.014 t } .$$ The mass, \(M _ { 2 }\) grams, of Substance 2 is increasing exponentially and the mass at certain times is shown in the following table. A critical stage in the experiment is reached at time \(T\) hours when the masses of the two substances are equal.

1. Find the rate at which the mass of Substance 1 is decreasing when \(t = 10\), giving your answer in grams per hour correct to 2 significant figures.
2. Show that \(T\) is the root of an equation of the form \(\mathrm { e } ^ { k t } = c\), where the values of the constants \(k\) and \(c\) are to be stated.
3. Hence find the value of \(T\) correct to 3 significant figures.

5

1. The mass, \(M\) grams, of a substance at time \(t\) years is given by $$M = 58 \mathrm { e } ^ { - 0.33 t }$$ Find the rate at which the mass is decreasing at the instant when \(t = 4\). Give your answer correct to 2 significant figures.
2. The mass of a second substance is increasing exponentially. The initial mass is 42.0 grams and, 6 years later, the mass is 51.8 grams. Find the mass at a time 24 years after the initial value.

3 The mass of a substance is decreasing exponentially. Its mass is \(m\) grams at time \(t\) years. The following table shows certain values of \(t\) and \(m\).

1. Find the values missing from the table.
2. Determine the value of \(t\), correct to the nearest integer, for which the mass is 50 grams.

2 The temperature \(T\) in degrees Celsius of water in a glass \(t\) minutes after boiling is modelled by the equation \(T = 20 + b \mathrm { e } ^ { - k t }\), where \(b\) and \(k\) are constants. Initially the temperature is \(100 ^ { \circ } \mathrm { C }\), and after 5 minutes the temperature is \(60 ^ { \circ } \mathrm { C }\).

1. Find \(b\) and \(k\).
2. Find at what time the temperature reaches \(50 ^ { \circ } \mathrm { C }\).

2 A radioactive substance decays exponentially, so that its mass \(M\) grams can be modelled by the equation \(M = A \mathrm { e } ^ { - k t }\), where \(t\) is the time in years, and \(A\) and \(k\) are positive constants.

1. An initial mass of 100 grams of the substance decays to 50 grams in 1500 years. Find \(A\) and \(k\).
2. The substance becomes safe when \(99 \%\) of its initial mass has decayed. Find how long it will take before the substance becomes safe.

5 A termites' nest has a population of \(P\) million. \(P\) is modelled by the equation \(P = 7 - 2 \mathrm { e } ^ { - k t }\), where \(t\) is in years, and \(k\) is a positive constant.

1. Calculate the population when \(t = 0\), and the long-term population, given by this model.
2. Given that the population when \(t = 1\) is estimated to be 5.5 million, calculate the value of \(k\).

7 Scientists can estimate the time elapsed since an animal died by measuring its body temperature.

1. Assuming the temperature goes down at a constant rate of 1.5 degrees Fahrenheit per hour, estimate how long it will take for the temperature to drop
   (A) from \(98 ^ { \circ } \mathrm { F }\) to \(89 ^ { \circ } \mathrm { F }\),
   (B) from \(98 ^ { \circ } \mathrm { F }\) to \(80 ^ { \circ } \mathrm { F }\). In practice, rate of temperature loss is not likely to be constant. A better model is provided by Newton's law of cooling, which states that the temperature \(\theta\) in degrees Fahrenheit \(t\) hours after death is given by the differential equation $$\frac { \mathrm { d } \theta } { \mathrm {~d} t } = - k \left( \theta - \theta _ { 0 } \right)$$ where \(\theta _ { 0 } { } ^ { \circ } \mathrm { F }\) is the air temperature and \(k\) is a constant.
2. Show by integration that the solution of this equation is \(\theta = \theta _ { 0 } + A \mathrm { e } ^ { - k t }\), where \(A\) is a constant. The value of \(\theta _ { 0 }\) is 50 , and the initial value of \(\theta\) is 98 . The initial rate of temperature loss is \(1.5 ^ { \circ } \mathrm { F }\) per hour.
3. Find \(A\), and show that \(k = 0.03125\).
4. Use this model to calculate how long it will take for the temperature to drop
   (A) from \(98 ^ { \circ } \mathrm { F }\) to \(89 ^ { \circ } \mathrm { F }\),
   (B) from \(98 ^ { \circ } \mathrm { F }\) to \(80 ^ { \circ } \mathrm { F }\).
5. Comment on the results obtained in parts (i) and (iv).

7 A drug is administered by an intravenous drip. The concentration, \(x\), of the drug in the blood is measured as a fraction of its maximum level. The drug concentration after \(t\) hours is modelled by the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = k \left( 1 + x - 2 x ^ { 2 } \right) ,$$ where \(0 \leqslant x < 1\), and \(k\) is a positive constant. Initially, \(x = 0\).

1. Express \(\frac { 1 } { ( 1 + 2 x ) ( 1 - x ) }\) in partial fractions.
2. Hence solve the differential equation to show that \(\frac { 1 + 2 x } { 1 - x } = \mathrm { e } ^ { 3 k t }\).
3. After 1 hour the drug concentration reaches \(75 \%\) of its maximum value and so \(x = 0.75\). Find the value of \(k\), and the time taken for the drug concentration to reach \(90 \%\) of its maximum value.
4. Rearrange the equation in part (ii) to show that \(x = \frac { 1 - \mathrm { e } ^ { - 3 k t } } { 1 + 2 \mathrm { e } ^ { - 3 k t } }\). Verify that in the long term the drug concentration approaches its maximum value. \section*{END OF QUESTION PAPER} \section*{Tuesday 16 J une 2015 - Afternoon} \section*{A2 GCE MATHEMATICS (MEI)} 4754/01B Applications of Advanced Mathematics (C4) Paper B: Comprehension \section*{QUESTION PAPER} \section*{Candidates answer on the Question Paper.} \section*{OCR supplied materials:}
   • Insert (inserted)
   • MEI Examination Formulae and Tables (MF2)
   \section*{Other materials required:}
   • Scientific or graphical calculator
   • Rough paper
   Duration: Up to 1 hour \includegraphics[max width=\textwidth, alt={}, center]{132ae754-bd4c-4819-80ef-4823ac2ead4f-05_117_495_1014_1308} PLEASE DO NOT WRITE IN THIS SPACE 2 In line 79 it says "For most journeys, more than half the journey time is composed of load time and transfer time". For what percentage of the journey time for the round trip made by car A in Table 4 is the car stationary?
   \includegraphics[max width=\textwidth, alt={}]{132ae754-bd4c-4819-80ef-4823ac2ead4f-07_645_1746_388_164}
   3 Using the expression on line 51, work out the answer to the question on lines 39 and 40 for the case where there are 10 upper floors and 7 people. Give your answer to 2 decimal places.
   \includegraphics[max width=\textwidth, alt={}]{132ae754-bd4c-4819-80ef-4823ac2ead4f-07_488_1746_1233_164}
   4 In lines 89 and 90 it says "... on average there will be approximately 8 stops per trip. A round trip with 8 stops would take between 188 and 200 seconds". Explain how the figure of 188 seconds has been derived. 5
5. Referring to Strategy 3 and lines 99 to 101, complete the table below for car C .
6. Calculate the time car C will take to transport all the people who work on floors 7 and 8 , and return to the ground floor.

   5	\includegraphics[max width=\textwidth, alt={}]{132ae754-bd4c-4819-80ef-4823ac2ead4f-08_1095_816_484_700}

   68 people make independent visits to any one of the upper floors of a building with 10 upper floors. What is the probability that at least one of the visitors goes to the top floor?

   6

   7 On lines 94 and 95 it says "Table 4 gives the timings for round trips in which the cars are required to stop at every floor they serve; Table 2 suggests this is a common occurrence in this case". Explain how Table 2 is used to make this claim. \includegraphics[max width=\textwidth, alt={}, center]{132ae754-bd4c-4819-80ef-4823ac2ead4f-09_1093_1740_1238_166} END OF QUESTION PAPER

11 In a science experiment a substance is decaying exponentially. Its mass, \(M\) grams, at time \(t\) minutes is given by \(M = 300 e ^ { - 0.05 t }\).

1. Find the time taken for the mass to decrease to half of its original value. A second substance is also decaying exponentially. Initially its mass was 400 grams and, after 10 minutes, its mass was 320 grams.
2. Find the time at which both substances are decaying at the same rate.

8

1. Substance \(A\) is decaying exponentially such that its mass is \(m\) grams at time \(t\) minutes. Find the missing values of \(m\) and \(t\) in the following table.

   \(t\)	0	10		50
   \(m\)	1250	750	450

2. Substance \(B\) is also decaying exponentially, according to the model \(m = 160 \mathrm { e } ^ { - 0.055 t }\), where \(m\) grams is its mass after \(t\) minutes.
   1. Determine the value of \(t\) for which the mass of substance \(B\) is half of its original mass.
   2. Determine the rate of decay of substance \(B\) when \(t = 15\).
3. State whether substance \(A\) or substance \(B\) is decaying at a faster rate, giving a reason for your answer.

11 The owners of an online shop believe that their sales can be modelled by \(S = a b ^ { t }\), where \(a\) and \(b\) are both positive constants, \(S\) is the number of items sold in a month and \(t\) is the number of complete months since starting their online shop. The sales for the first six months are recorded, and the values of \(\log _ { 10 } S\) are plotted against \(t\) in the graph below. The graph is repeated in the Printed Answer Booklet. \includegraphics[max width=\textwidth, alt={}, center]{9473b8f7-616a-485e-963b-696c6640ae6b-08_1203_1408_552_244}

1. Explain why the graph suggests that the given model is appropriate. The owners believe that \(a = 120\) and \(b = 1.15\) are good estimates for the parameters in the model.
2. Show that the graph supports these estimates for the parameters.
3. Use the model \(S = 120 \times 1.15 ^ { t }\) to predict the number of items sold in the seventh month after opening.
4. 1. Use the model \(S = 120 \times 1.15 ^ { t }\) to predict the number of months after opening when the total number of items sold after opening will first exceed 70000 .
   2. Comment on how reliable this prediction may be.

4 A species of animal is to be introduced onto a remote island. Their food will consist only of various plants that grow on the island. A zoologist proposes two possible models for estimating the population \(P\) after \(t\) years. The diagrams show these models as they apply to the first 20 years. \includegraphics[max width=\textwidth, alt={}, center]{1a0e0afb-81be-45d1-8c86-f98e508e9a49-05_725_606_406_242} \includegraphics[max width=\textwidth, alt={}, center]{1a0e0afb-81be-45d1-8c86-f98e508e9a49-05_714_593_413_968}

1. Without calculation, describe briefly how the rate of growth of \(P\) will vary for the first 20 years, according to each of these two models. The equation of the curve for model A is \(P = 20 + 1000 \mathrm { e } ^ { - \frac { ( t - 20 ) ^ { 2 } } { 100 } }\).
   The equation of the curve for model B is \(P = 20 + 1000 \left( 1 - \mathrm { e } ^ { - \frac { t } { 5 } } \right)\).
2. Describe the behaviour of \(P\) that is predicted for \(t > 20\)
   1. using model A,
   2. using model B . There is only a limited amount of food available on the island, and the zoologist assumes that the size of the population depends on the amount of food available and on no other external factors.
3. State what is suggested about the long-term food supply by
   1. model A,
   2. model B.

6 Helga invests \(\pounds 4000\) in a savings account.
After \(t\) days, her investment is worth \(\pounds y\).
The rate of increase of \(y\) is \(k y\), where \(k\) is a constant.

1. Write down a differential equation in terms of \(t , y\) and \(k\).
2. Solve your differential equation to find the value of Helga's investment after \(t\) days. Give your answer in terms of \(k\) and \(t\). It is given that \(k = \frac { 1 } { 365 } \ln \left( 1 + \frac { r } { 100 } \right)\) where \(r \%\) is the rate of interest per annum. During the first year the rate of interest is \(6 \%\) per annum.
3. Find the value of Helga's investment after 90 days. After one year (365 days), the rate of interest drops to 5\% per annum.
4. Find the total time that it will take for Helga's investment to double in value.

1. The value of a car, \(\pounds V\), can be modelled by the equation

$$V = 15700 \mathrm { e } ^ { - 0.25 t } + 2300 \quad t \in \mathbb { R } , t \geqslant 0$$ where the age of the car is \(t\) years.
Using the model,

1. find the initial value of the car. Given the model predicts that the value of the car is decreasing at a rate of \(\pounds 500\) per year at the instant when \(t = T\),
2. 1. show that $$3925 \mathrm { e } ^ { - 0.25 T } = 500$$
   2. Hence find the age of the car at this instant, giving your answer in years and months to the nearest month.
      (Solutions based entirely on graphical or numerical methods are not acceptable.) The model predicts that the value of the car approaches, but does not fall below, \(\pounds A\).
3. State the value of \(A\).
4. State a limitation of this model.

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.

1. The mass, \(A\) kg, of algae in a small pond, is modelled by the equation

$$A = p q ^ { t }$$ where \(p\) and \(q\) are constants and \(t\) is the number of weeks after the mass of algae was first recorded. Data recorded indicates that there is a linear relationship between \(t\) and \(\log _ { 10 } A\) given by the equation $$\log _ { 10 } A = 0.03 t + 0.5$$

1. Use this relationship to find a complete equation for the model in the form $$A = p q ^ { t }$$ giving the value of \(p\) and the value of \(q\) each to 4 significant figures.
2. With reference to the model, interpret
   1. the value of the constant \(p\),
   2. the value of the constant \(q\).
3. Find, according to the model,
   1. the mass of algae in the pond when \(t = 8\), giving your answer to the nearest 0.5 kg ,
   2. the number of weeks it takes for the mass of algae in the pond to reach 4 kg .
4. State one reason why this may not be a realistic model in the long term.

7. \begin{figure}[h] \end{figure} A chimney emits smoke particles.
On a particular day, the concentration of smoke particles in the air emitted by this chimney, \(P\) parts per million, is measured at various distances, \(x \mathrm {~km}\), from the chimney. Figure 2 shows a sketch of the linear relationship between \(\log _ { 10 } P\) and \(x\) that is used to model this situation. The line passes through the point ( \(0,3.3\) ) and the point ( \(6,2.1\) )

1. Find a complete equation for the model in the form $$P = a b ^ { x }$$ where \(a\) and \(b\) are constants. Give the value of \(a\) and the value of \(b\) each to 4 significant figures.
2. With reference to the model, interpret the value of \(a b\)

1. The prices of two precious metals are being monitored.

The price per gram of metal \(A , \pounds V _ { A }\), is modelled by the equation $$V _ { A } = 100 + 20 \mathrm { e } ^ { 0.04 t }$$ where \(t\) is the number of months after monitoring began.
The price per gram of metal \(B , \pounds V _ { B }\), is modelled by the equation $$V _ { B } = p \mathrm { e } ^ { - 0.02 t }$$ where \(p\) is a positive constant and \(t\) is the number of months after monitoring began.
Given that \(V _ { B } = 2 V _ { A }\) when \(t = 0\)

1. find the value of \(p\) When \(t = T\), the rate of increase in the price per gram of metal \(A\) was equal to the rate of decrease in the price per gram of metal \(B\)
2. Find the value of \(T\), giving your answer to one decimal place.
   (Solutions based entirely on calculator technology are not acceptable.)

1. The owners of a nature reserve decided to increase the area of the reserve covered by trees.

Tree planting started on 1st January 2005.
The area of the nature reserve covered by trees, \(A \mathrm {~km} ^ { 2 }\), is modelled by the equation $$A = 80 - 45 \mathrm { e } ^ { c t }$$ where \(c\) is a constant and \(t\) is the number of years after 1st January 2005.
Using the model,

1. find the area of the nature reserve that was covered by trees just before tree planting started. On 1st January 2019 an area of \(60 \mathrm {~km} ^ { 2 }\) of the nature reserve was covered by trees.
2. Use this information to find a complete equation for the model, giving your value of \(c\) to 3 significant figures. On 1st January 2020, the owners of the nature reserve announced a long-term plan to have \(100 \mathrm {~km} ^ { 2 }\) of the nature reserve covered by trees.
3. State a reason why the model is not appropriate for this plan.

1. The growth of pond weed on the surface of a pond is being investigated.

The surface area of the pond covered by the weed, \(A \mathrm {~m} ^ { 2 }\), can be modelled by the equation $$A = 0.2 \mathrm { e } ^ { 0.3 t }$$ where \(t\) is the number of days after the start of the investigation.

1. State the surface area of the pond covered by the weed at the start of the investigation.
2. Find the rate of increase of the surface area of the pond covered by the weed, in \(\mathrm { m } ^ { 2 } /\) day, exactly 5 days after the start of the investigation. Given that the pond has a surface area of \(100 \mathrm {~m} ^ { 2 }\),
3. find, to the nearest hour, the time taken, according to the model, for the surface of the pond to be fully covered by the weed. The pond is observed for one month and by the end of the month \(90 \%\) of the surface area of the pond was covered by the weed.
4. Evaluate the model in light of this information, giving a reason for your answer.