1.06i Exponential growth/decay: in modelling context

1. The value of Bob's car can be calculated from the formula

$$V = 17000 \mathrm { e } ^ { - 0.25 t } + 2000 \mathrm { e } ^ { - 0.5 t } + 500$$ where \(V\) is the value of the car in pounds \(( \pounds )\) and \(t\) is the age in years.

1. Find the value of the car when \(t = 0\)
2. Calculate the exact value of \(t\) when \(V = 9500\)
3. Find the rate at which the value of the car is decreasing at the instant when \(t = 8\). Give your answer in pounds per year to the nearest pound.

1. A pot of coffee is delivered to a meeting room at 11 am . At a time \(t\) minutes after 11 am the temperature, \(\theta ^ { \circ } \mathrm { C }\), of the coffee in the pot is given by the equation

$$\theta = A + 60 \mathrm { e } ^ { - k t }$$ where \(A\) and \(k\) are positive constants. Given also that the temperature of the coffee at 11 am is \(85 ^ { \circ } \mathrm { C }\) and that 15 minutes later it is \(58 ^ { \circ } \mathrm { C }\),

1. find the value of \(A\).
2. Show that \(k = \frac { 1 } { 15 } \ln \left( \frac { 20 } { 11 } \right)\)
3. Find, to the nearest minute, the time at which the temperature of the coffee reaches \(50 ^ { \circ } \mathrm { C }\).

1. 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.

1. A heated metal ball is dropped into a liquid. As the ball cools, its temperature, \(T ^ { \circ } \mathrm { C }\), \(t\) minutes after it enters the liquid, is given by

$$T = 400 \mathrm { e } ^ { - 0.05 t } + 25 , \quad t \geqslant 0$$

1. Find the temperature of the ball as it enters the liquid.
2. Find the value of \(t\) for which \(T = 300\), giving your answer to 3 significant figures.
3. Find the rate at which the temperature of the ball is decreasing at the instant when \(t = 50\). Give your answer in \({ } ^ { \circ } \mathrm { C }\) per minute to 3 significant figures.
4. From the equation for temperature \(T\) in terms of \(t\), given above, explain why the temperature of the ball can never fall to \(20 ^ { \circ } \mathrm { C }\).

8. The amount of a certain type of drug in the bloodstream \(t\) hours after it has been taken is given by the formula $$x = D \mathrm { e } ^ { - \frac { 1 } { 8 } t } ,$$ where \(x\) is the amount of the drug in the bloodstream in milligrams and \(D\) is the dose given in milligrams. A dose of 10 mg of the drug is given.

1. Find the amount of the drug in the bloodstream 5 hours after the dose is given. Give your answer in mg to 3 decimal places. A second dose of 10 mg is given after 5 hours.
2. Show that the amount of the drug in the bloodstream 1 hour after the second dose is 13.549 mg to 3 decimal places. No more doses of the drug are given. At time \(T\) hours after the second dose is given, the amount of the drug in the bloodstream is 3 mg .
3. Find the value of \(T\).

1. Rabbits were introduced onto an island. The number of rabbits, \(P , t\) years after they were introduced is modelled by the equation

$$P = 80 \mathrm { e } ^ { \frac { 1 } { 5 } t } , \quad t \in \mathbb { R } , t \geqslant 0$$

1. Write down the number of rabbits that were introduced to the island.
2. Find the number of years it would take for the number of rabbits to first exceed 1000.
3. Find \(\frac { \mathrm { d } P } { \mathrm {~d} t }\).
4. Find \(P\) when \(\frac { \mathrm { d } P } { \mathrm {~d} t } = 50\).

5. The mass, \(m\) grams, of a leaf \(t\) days after it has been picked from a tree is given by $$m = p \mathrm { e } ^ { - k t }$$ where \(k\) and \(p\) are positive constants.
When the leaf is picked from the tree, its mass is 7.5 grams and 4 days later its mass is 2.5 grams.

1. Write down the value of \(p\).
2. Show that \(k = \frac { 1 } { 4 } \ln 3\).
3. Find the value of \(t\) when \(\frac { \mathrm { d } m } { \mathrm {~d} t } = - 0.6 \ln 3\).

8. \begin{figure}[h] \end{figure} The population of a town is being studied. The population \(P\), at time \(t\) years from the start of the study, is assumed to be $$P = \frac { 8000 } { 1 + 7 \mathrm { e } ^ { - k t } } , \quad t \geqslant 0$$ where \(k\) is a positive constant.
The graph of \(P\) against \(t\) is shown in Figure 3. Use the given equation to

1. find the population at the start of the study,
2. find a value for the expected upper limit of the population. Given also that the population reaches 2500 at 3 years from the start of the study,
3. calculate the value of \(k\) to 3 decimal places. Using this value for \(k\),
4. find the population at 10 years from the start of the study, giving your answer to 3 significant figures.
5. Find, using \(\frac { \mathrm { d } P } { \mathrm {~d} t }\), the rate at which the population is growing at 10 years from the start of the study.

9. The amount of an antibiotic in the bloodstream, from a given dose, is modelled by the formula $$x = D \mathrm { e } ^ { - 0.2 t }$$ where \(x\) is the amount of the antibiotic in the bloodstream in milligrams, \(D\) is the dose given in milligrams and \(t\) is the time in hours after the antibiotic has been given. A first dose of 15 mg of the antibiotic is given.

1. Use the model to find the amount of the antibiotic in the bloodstream 4 hours after the dose is given. Give your answer in mg to 3 decimal places. A second dose of 15 mg is given 5 hours after the first dose has been given. Using the same model for the second dose,
2. show that the total amount of the antibiotic in the bloodstream 2 hours after the second dose is given is 13.754 mg to 3 decimal places. No more doses of the antibiotic are given. At time \(T\) hours after the second dose is given, the total amount of the antibiotic in the bloodstream is 7.5 mg .
3. Show that \(T = a \ln \left( b + \frac { b } { \mathrm { e } } \right)\), where \(a\) and \(b\) are integers to be determined.
   

   VIIIV SIHI NITIIIM I I N O C

   VI4V SIHI NI IHIHM ION OC

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   \includegraphics[max width=\textwidth, alt={}, center]{d3ba2776-eedb-48f0-834f-41aa454afba3-16_2258_47_315_37}

1. The value of a car is modelled by the formula

$$V = 16000 \mathrm { e } ^ { - k t } + A , \quad t \geqslant 0 , t \in \mathbb { R }$$ where \(V\) is the value of the car in pounds, \(t\) is the age of the car in years, and \(k\) and \(A\) are positive constants. Given that the value of the car is \(\pounds 17500\) when new and \(\pounds 13500\) two years later,

1. find the value of \(A\),
2. show that \(k = \ln \left( \frac { 2 } { \sqrt { 3 } } \right)\)
3. Find the age of the car, in years, when the value of the car is \(\pounds 6000\) Give your answer to 2 decimal places.

3 The profit \(\pounds P\) made by a company in its \(n\)th year is modelled by the exponential function $$P = A \mathrm { e } ^ { b n }$$ In the first year (when \(n = 1\) ), the profit was \(\pounds 10000\). In the second year, the profit was \(\pounds 16000\).

1. Show that \(\mathrm { e } ^ { b } = 1.6\), and find \(b\) and \(A\).
2. What does this model predict the profit to be in the 20th year?

11 Answer part (iii) of this question on the insert provided.
A hot drink is made and left to cool. The table shows its temperature at ten-minute intervals after it is made. The room temperature is \(22 ^ { \circ } \mathrm { C }\). The difference between the temperature of the drink and room temperature at time \(t\) minutes is \(z ^ { \circ } \mathrm { C }\). The relationship between \(z\) and \(t\) is modelled by $$z = z _ { 0 } 10 ^ { - k t }$$ where \(z _ { 0 }\) and \(k\) are positive constants.

1. Give a physical interpretation for the constant \(z _ { 0 }\).
2. Show that \(\log _ { 10 } z = - k t + \log _ { 10 } z _ { 0 }\).
3. On the insert, complete the table and draw the graph of \(\log _ { 10 } z\) against \(t\). Use your graph to estimate the values of \(k\) and \(z _ { 0 }\).
   Hence estimate the temperature of the drink 70 minutes after it is made.

10 A culture of bacteria is observed during an experiment. The number of bacteria is denoted by \(N\) and the time in hours after the start of the experiment by \(t\).
The table gives observations of \(t\) and \(N\).

1. Plot the points \(( t , N )\) on graph paper and join them with a smooth curve.
2. Explain why the curve suggests why the relationship connecting \(t\) and \(N\) may be of the form \(N = a b ^ { t }\).
3. Explain how, by using logarithms, the curve given by plotting \(N\) against \(t\) can be transformed into a straight line.
   State the gradient of this straight line and its intercept with the vertical axis in terms of \(a\) and \(b\).
4. Complete a table of values for \(\log _ { 10 } N\) and plot the points \(\left( t , \log _ { 10 } N \right)\) on graph paper. Draw the best fit line through the points and use it to estimate the values of \(a\) and \(b\).

2 The growth in population \(P\) of a certain town after time \(t\) years can be modelled by the equation \(P = 11000 \times 10 ^ { k t }\) where \(k\) is a constant.

1. State the initial population of the town.
2. After three years the population of the town is 24000 . Use this information to find the value of \(k\) correct to two decimal places.

8. A rock contains a radioactive substance which is decaying. The mass of the rock, \(m\) grams, at time \(t\) years after initial observation is given by $$m = 400 + 80 \mathrm { e } ^ { - k t }$$ where \(k\) is a positive constant.
Given that the mass of the rock decreases by \(0.2 \%\) in the first 10 years, find

1. the value of \(k\),
2. the value of \(t\) when \(m = 475\),
3. the rate at which the mass of the rock is decreasing when \(t = 100\).

9. \includegraphics[max width=\textwidth, alt={}, center]{d1cf3850-964a-4ff1-ae25-f1bc60a6aded-3_501_1111_877_413} The diagram 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 \(70 \leq t \leq 120\) $$T = 5 + B \mathrm { e } ^ { - k t }$$

9. 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 \(t = 3\).

6

1. \(t\)	0	10	20
   \(X\)	275	440

   The quantity \(X\) is increasing exponentially with respect to time \(t\). The table above shows values of \(X\) for different values of \(t\). Find the value of \(X\) when \(t = 20\).
2. The quantity \(Y\) is decreasing exponentially with respect to time \(t\) where $$Y = 80 \mathrm { e } ^ { - 0.02 t }$$
   1. Find the value of \(t\) for which \(Y = 20\), giving your answer correct to 2 significant figures.
   2. Find by differentiation the rate at which \(Y\) is decreasing when \(t = 30\), giving your answer correct to 2 significant figures.

3 The mass, \(m\) grams, of a substance at time \(t\) years is given by the formula $$m = 180 \mathrm { e } ^ { - 0.017 t } .$$

1. Find the value of \(t\) for which the mass is 25 grams.
2. Find the rate at which the mass is decreasing when \(t = 55\).

5 A substance is decaying in such a way that its mass, m kg , at a time t years from now is given by the formula $$\mathrm { m } = 240 \mathrm { e } ^ { - 0.04 \mathrm { t } }$$

1. Find the time taken for the substance to halve its mass.
2. Find the value of \(t\) for which the mass is decreasing at a rate of 2.1 kg per year.

7 It is claimed that the number of plants of a certain species in a particular locality is doubling every 9 years. The number of plants now is 42 . The number of plants is treated as a continuous variable and is denoted by \(N\). The number of years from now is denoted by \(t\).

1. Two equivalent expressions giving \(N\) in terms of \(t\) are $$N = A \times 2 ^ { k t } \quad \text { and } \quad N = A \mathrm { e } ^ { m t } .$$ Determine the value of each of the constants \(A , k\) and \(m\).
2. Find the value of \(t\) for which \(N = 100\), giving your answer correct to 3 significant figures.
3. Find the rate at which the number of plants will be increasing at a time 35 years from now.

5 At time \(t\) minutes after an oven is switched on, its temperature \(\theta ^ { \circ } \mathrm { C }\) is given by $$\theta = 200 - 180 \mathrm { e } ^ { - 0.1 t }$$

1. State the value which the oven's temperature approaches after a long time.
2. Find the time taken for the oven's temperature to reach \(150 ^ { \circ } \mathrm { C }\).
3. Find the rate at which the temperature is increasing at the instant when the temperature reaches \(150 ^ { \circ } \mathrm { C }\).

2 A population is \(P\) million at time \(t\) years. \(P\) is modelled by the equation $$P = 5 + a \mathrm { e } ^ { - b t }$$ where \(a\) and \(b\) are constants.
The population is initially 8 million, and declines to 6 million after 1 year.

1. Use this information to calculate the values of \(a\) and \(b\), giving \(b\) correct to 3 significant figures.
2. What is the long-term population predicted by the model?

3 The value \(\pounds V\) of a car is modelled by the equation \(V = A \mathrm { e } ^ { - k t }\), where \(t\) is the age of the car in years and \(A\) and \(k\) are constants. Its value when new is \(\pounds 10000\), and after 3 years its value is \(\pounds 6000\).

1. Find the values of \(A\) and \(k\).
2. Find the age of the car when its value is \(\pounds 2000\).

6 The mass \(M \mathrm {~kg}\) of a radioactive material is modelled by the equation $$M = M _ { 0 } \mathrm { e } ^ { - k t } ,$$ where \(M _ { 0 }\) is the initial mass, \(t\) is the time in years, and \(k\) is a constant which measures the rate of radioactive decay.

1. Sketch the graph of \(M\) against \(t\).
2. For Carbon \(14 , k = 0.000121\). Verify that after 5730 years the mass \(M\) has reduced to approximately half the initial mass. The half-life of a radioactive material is the time taken for its mass to reduce to exactly half the initial mass.
3. Show that, in general, the half-life \(T\) is given by \(T = \frac { \ln 2 } { k }\).
4. Hence find the half-life of Plutonium 239, given that for this material \(k = 2.88 \times 10 ^ { - 5 }\).