Exponential growth/decay model setup

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. 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 \(t\).

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 \(\frac { \mathrm { d } P } { \mathrm {~d} t } = 50\).

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

   VI4V SIHI NI JIIIM ION OO

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

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?

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.

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.

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.

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 \(\pounds 2000\).

4 The graph of \(y = a b ^ { x }\) passes through the points \(( 1,6 )\) and \(( 2,3.6 )\). Find the values of \(a\) and \(b\).

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,
   (a) find the values missing from the table,
   (b) determine the value of \(k\).
2. 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 $$M = 40 \mathrm { e } ^ { - 0.132 t }$$ (a) Find the time taken for the mass of substance \(A\) to decrease to \(25 \%\) of its value when \(t = 0\).
   (b) Find the rate at which the mass of substance \(A\) is decreasing when \(t = 5\).
2. 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.

6 The value \(\pounds V\) of a car \(t\) years after it is new is modelled by the equation \(V = A \mathrm { e } ^ { - k t }\), where \(A\) and \(k\) are positive constants which depend on the make and model of the car.

1. Brian buys a new sports car. Its value is modelled by the equation $$V = 20000 \mathrm { e } ^ { - 0.2 t } .$$ Calculate how much value, to the nearest \(\pounds 100\), this car has lost after 1 year.
2. At the same time as Brian buys his car, Kate buys a new hatchback for \(\pounds 15000\). Her car loses \(\pounds 2000\) of its value in the first year. Show that, for Kate's car, \(k = 0.143\) correct to 3 significant figures.
3. Find how long it is before Brian's and Kate's cars have the same value.

3 It is given that \(x\) is proportional to the product of the square of \(y\) and the positive square root of \(z\). When \(y = 2\) and \(z = 9 , x = 30\).

1. Write an equation for \(x\) in terms of \(y\) and \(z\).
2. Find the value of \(x\) when \(y = 3\) and \(z = 25\).

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.

1. A tree was planted in the ground.

Its height, \(H\) metres, was measured \(t\) years after planting.
Exactly 3 years after planting, the height of the tree was 2.35 metres.
Exactly 6 years after planting, the height of the tree was 3.28 metres.
Using a linear model,

1. find an equation linking \(H\) with \(t\). The height of the tree was approximately 140 cm when it was planted.
2. Explain whether or not this fact supports the use of the linear model in part (a).

1. A scientist is studying the growth of 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 1000 bacteria in this population at the start of the study
• it took exactly 5 hours from the start of the study for this population to double
  1. find a complete equation for the model.
  2. Hence find the rate of increase in the number of bacteria in this population exactly 8 hours from the start of the study. Give your answer to 2 significant figures.

The number of bacteria, \(M\), in the second population is modelled by the equation $$M = 500 \mathrm { e } ^ { 1.4 k t } \quad t \geqslant 0$$ where \(k\) has the value found in part (a) and \(t\) is the time in hours from the start of the study.
Given that \(T\) hours after the start of the study, the number of bacteria in the two different populations was the same,

find the value of \(T\).

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 The power output, \(P\) watts, of a certain wind turbine is proportional to the cube of the wind speed \(v \mathrm {~ms} ^ { - 1 }\). When \(v = 3.6 , P = 50\).
Determine the wind speed that will give a power output of 225 watts.