Finding x from given y value

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.

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. The population in thousands, \(P\), of a town at time \(t\) years after \(1 ^ { \text {st } }\) January 1980 is modelled by the formula

$$P = 30 + 50 \mathrm { e } ^ { 0.002 t }$$ Use this model to estimate

1. the population of the town on \(1 { } ^ { \text {st } }\) January 2010,
2. the year in which the population first exceeds 84000 . The population in thousands, \(Q\), of another town is modelled by the formula $$Q = 26 + 50 \mathrm { e } ^ { 0.003 t }$$
3. Show that the value of \(t\) when \(P = Q\) is a solution of the equation $$t = 1000 \ln \left( 1 + 0.08 \mathrm { e } ^ { - 0.002 t } \right) .$$
4. Use the iteration formula $$t _ { n + 1 } = 1000 \ln \left( 1 + 0.08 \mathrm { e } ^ { - 0.002 t _ { n } } \right)$$ with \(t _ { 0 } = 50\) to find \(t _ { 1 } , t _ { 2 }\) and \(t _ { 3 }\) and hence, the year in which the populations of these two towns will be equal according to these models.

5. 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 \(t = 3\).

4 A scientist is testing models for the growth and decay of colonies of bacteria. For a particular colony, which is growing, the model is \(P = A \mathrm { e } ^ { \frac { 1 } { 8 } t }\), where \(P\) is the number of bacteria after a time \(t\) minutes and \(A\) is a constant.

1. This growing colony consists initially of 500 bacteria. Calculate the number of bacteria, according to the model, after one hour. Give your answer to the nearest thousand.
2. For a second colony, which is decaying, the model is \(Q = 500000 \mathrm { e } ^ { - \frac { 1 } { 8 } t }\), where \(Q\) is the number of bacteria after a time \(t\) minutes. Initially, the growing colony has 500 bacteria and, at the same time, the decaying colony has 500000 bacteria.
   1. Find the time at which the populations of the two colonies will be equal, giving your answer to the nearest 0.1 of a minute.
   2. The population of the growing colony will exceed that of the decaying colony by 45000 bacteria at time \(T\) minutes. Show that $$\left( \mathrm { e } ^ { \frac { 1 } { 8 } T } \right) ^ { 2 } - 90 \mathrm { e } ^ { \frac { 1 } { 8 } T } - 1000 = 0$$ and hence find the value of \(T\), giving your answer to one decimal place.
      (4 marks)

4 A painting was valued on 1 April 2001 at \(\pounds 5000\).
The value of this painting is modelled by $$V = A p ^ { t }$$ where \(\pounds V\) is the value \(t\) years after 1 April 2001, and \(A\) and \(p\) are constants.

1. Write down the value of \(A\).
2. According to the model, the value of this painting on 1 April 2011 was \(\pounds 25000\). Using this model:
   1. show that \(p ^ { 10 } = 5\);
   2. use logarithms to find the year in which the painting will be valued at \(\pounds 75000\).
3. A painting by another artist was valued at \(\pounds 2500\) on 1 April 1991. The value of this painting is modelled by $$W = 2500 q ^ { t }$$ where \(\pounds W\) is the value \(t\) years after 1 April 1991, and \(q\) is a constant.
   1. Show that, according to the two models, the value of the two paintings will be the same \(T\) years after 1 April 1991, $$\text { where } T = \frac { \ln \left( \frac { 5 } { 2 } \right) } { \ln \left( \frac { p } { q } \right) }$$
   2. Given that \(p = 1.029 q\), find the year in which the two paintings will have the same value.
      [0pt] [1 mark]

6 An on-line science website states:
'To find a dog's equivalent human age in years, multiply the natural logarithm of the dog's age in years by 16 then add 31.' 6

1. Calculate the equivalent age to the nearest human year of a dog aged 5 years. 6
2. A dog's equivalent age in human years is 40 years. Find the dog's actual age to the nearest month.
   6
3. Explain why the behaviour of the natural logarithm for values close to zero means that the formula given on the website cannot be true for very young dogs.

During one day, a biological culture 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]
2. find, to the nearest minute, the time when the initial number of bacteria will have doubled. [4]

The temperature \(\theta\) °C of water in a container after \(t\) minutes is modelled by the equation $$\theta = a - be^{-kt},$$ where \(a\), \(b\) and \(k\) are positive constants. The initial and long-term temperatures of the water are 15°C and 100°C respectively. After 1 minute, the temperature is 30°C.

1. Find \(a\), \(b\) and \(k\). [6]
2. Find how long it takes for the temperature to reach 80°C. [2]

The height \(h\) metres of a tree after \(t\) years is modelled by the equation $$h = a - be^{-kt},$$ where \(a\), \(b\) and \(k\) are positive constants.

1. Given that the long-term height of the tree is 10.5 metres, and the initial height is 0.5 metres, find the values of \(a\) and \(b\). [3]
2. Given also that the tree grows to a height of 6 metres in 8 years, find the value of \(k\), giving your answer correct to 2 decimal places. [3]

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

The mass of radioactive atoms in a substance can be modelled by the equation $$m = m_0 k^t$$ where \(m_0\) grams is the initial mass, \(m\) grams is the mass after \(t\) days and \(k\) is a constant. The value of \(k\) differs from one substance to another.

1. 1. A sample of radioactive iodine reduced in mass from 24 grams to 12 grams in 8 days. Show that the value of the constant \(k\) for this substance is 0.917004, correct to six decimal places. [1 mark]
   2. A similar sample of radioactive iodine reduced in mass to 1 gram after 60 days. Calculate the initial mass of this sample, giving your answer to the nearest gram. [2 marks]
2. The half-life of a radioactive substance is the time it takes for a mass of \(m_0\) to reduce to a mass of \(\frac{1}{2}m_0\). A sample of radioactive vanadium reduced in mass from exactly 10 grams to 8.106 grams in 100 days. Find the half-life of radioactive vanadium, giving your answer to the nearest day. [4 marks]

Charlie buys a car for £18000 on 1 January 2016. The value of the car decreases exponentially. The car has a value of £12000 on 1 January 2018.

1. Charlie says: • because the car has lost £6000 after two years, after another two years it will be worth £6000. Charlie's friend Kaya says: • because the car has lost one third of its value after two years, after another two years it will be worth £8000. Explain whose statement is correct, justifying the value they have stated. [2 marks]
2. The value of Charlie's car, £\(V\), \(t\) years after 1 January 2016 may be modelled by the equation $$V = Ae^{-kt}$$ where \(A\) and \(k\) are positive constants. Find the value of \(t\) when the car has a value of £10000, giving your answer to two significant figures. [5 marks]
3. Give a reason why the model, in this context, will not be suitable to calculate the value of the car when \(t = 30\) [1 mark]

Trees in a forest may be affected by one of two types of fungal disease, but not by both. The number of trees affected by disease A, \(n_A\), can be modelled by the formula $$n_A = ae^{0.1t}$$ where \(t\) is the time in years after 1 January 2017. The number of trees affected by disease B, \(n_B\), can be modelled by the formula $$n_B = be^{0.2t}$$ On 1 January 2017 a total of 290 trees were affected by a fungal disease. On 1 January 2018 a total of 331 trees were affected by a fungal disease.

1. Show that \(b = 90\), to the nearest integer, and find the value of \(a\). [3 marks]
2. Estimate the total number of trees that will be affected by a fungal disease on 1 January 2020. [1 mark]
3. Find the year in which the number of trees affected by disease B will first exceed the number affected by disease A. [3 marks]
4. Comment on the long-term accuracy of the model. [1 mark]

David has been investigating the population of rabbits on an island during a three-year period. Based on data that he has collected, David decides to model the population of rabbits, \(R\), by the formula $$R = 50e^{0.5t}$$ where \(t\) is the time in years after 1 January 2016.

1. Using David's model:
   1. state the population of rabbits on the island on 1 January 2016; [1 mark]
   2. predict the population of rabbits on 1 January 2021. [1 mark]
2. Use David's model to find the value of \(t\) when \(R = 150\), giving your answer to three significant figures. [2 marks]
3. Give one reason why David's model may not be appropriate. [1 mark]
4. On the same island, the population of crickets, \(C\), can be modelled by the formula $$C = 1000e^{-0.1t}$$ where \(t\) is the time in years after 1 January 2016. Using the two models, find the year during which the population of rabbits first exceeds the population of crickets. [3 marks]

A student is conducting an experiment in a laboratory to investigate how quickly liquids cool to room temperature. A beaker containing a hot liquid at an initial temperature of \(75°C\) cools so that the temperature, \(\theta °C\), of the liquid at time \(t\) minutes can be modelled by the equation $$\theta = 5(4 + \lambda e^{-kt})$$ where \(\lambda\) and \(k\) are constants. After 2 minutes the temperature falls to \(68°C\).

1. Find the temperature of the liquid after 15 minutes. Give your answer to three significant figures. [7 marks]
2. 1. Find the room temperature of the laboratory, giving a reason for your answer. [2 marks]
   2. Find the time taken in minutes for the liquid to cool to \(1°C\) above the room temperature of the laboratory. [2 marks]
3. Explain why the model might need to be changed if the experiment was conducted in a different place. [1 mark]

The size \(N\) of the population of a small island at time \(t\) years may be modelled by \(N = Ae^{kt}\), where \(A\) and \(k\) are constants. It is known that \(N = 100\) when \(t = 2\) and that \(N = 160\) when \(t = 12\).

1. Interpret the constant \(A\) in the context of the question. [1]
2. Show that \(k = 0.047\), correct to three decimal places. [4]
3. Find the size of the population when \(t = 20\). [3]

A treatment is used to reduce the concentration of nitrate in the water in a pond. The concentration of nitrate in the pond water, \(N\) ppm (parts per million), is modelled by the equation $$N = 65 - 3e^{0.1t} \quad t \in \mathbb{R} \quad t \geq 0$$ where \(t\) hours is the time after the treatment was applied. Use the equation of the model to answer parts (a) and (b).

1. Calculate the reduction in the concentration of nitrate in the pond water in the first 8 hours after the treatment was applied. [3] For fish to survive in the pond, the concentration of nitrate in the water must be no more than 20 ppm.
2. Calculate the minimum time, after the treatment is applied, before fish can be safely introduced into the pond. Give your answer in hours to one decimal place. [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]
2. Determine the value of \(t\), correct to the nearest integer, for which the mass is 50 grams. [4]