Exponential model with shifted asymptote

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 \(5 ^ { \circ } \mathrm { C }\)

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

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.

8. \begin{figure}[h] \end{figure} The value of Lin's car is modelled by the formula $$V = 18000 \mathrm { e } ^ { - 0.2 t } + 4000 \mathrm { e } ^ { - 0.1 t } + 1000 , \quad t \geqslant 0$$ where the value of the car is \(V\) pounds when the age of the car is \(t\) years.
A sketch of \(t\) against \(V\) is shown in Figure 1.

1. State the range of \(V\). According to this model,
2. find the rate at which the value of the car is decreasing when \(t = 10\) Give your answer in pounds per year.
3. Calculate the exact value of \(t\) when \(V = 15000\)

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 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 \(20 ^ { \circ } \mathrm { C }\).

1. Water is being heated in an electric kettle. The temperature, \(\theta ^ { \circ } \mathrm { C }\), of the water \(t\) seconds after the kettle is switched on, is modelled by the equation

$$\theta = 120 - 100 \mathrm { e } ^ { - \lambda t } , \quad 0 \leqslant t \leqslant T$$

1. State the value of \(\theta\) when \(t = 0\) Given that the temperature of the water in the kettle is \(70 ^ { \circ } \mathrm { C }\) when \(t = 40\),
2. find the exact value of \(\lambda\), giving your answer in the form \(\frac { \ln a } { b }\), where \(a\) and \(b\) are integers. When \(t = T\), the temperature of the water reaches \(100 ^ { \circ } \mathrm { C }\) and the kettle switches off.
3. Calculate the value of \(T\) to the nearest whole number.

6. As a substance cools its temperature, \(T ^ { \circ } \mathrm { C }\), is related to the time ( \(t\) minutes) for which it has been cooling. The relationship is given by the equation $$T = 20 + 60 \mathrm { e } ^ { - 0.1 t } , t \geq 0$$

1. Find the value of \(T\) when the substance started to cool.
2. Explain why the temperature of the substance is always above \(20 ^ { \circ } \mathrm { C }\).
3. Sketch the graph of \(T\) against \(t\).
4. Find the value, to 2 significant figures, of \(t\) at the instant \(T = 60\).
5. Find \(\frac { \mathrm { d } T } { \mathrm {~d} t }\).
6. Hence find the value of \(T\) at which the temperature is decreasing at a rate of \(1.8 ^ { \circ } \mathrm { C }\) per minute.

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 \(70 \leq t \leq 120\) $$T = 5 + B \mathrm { e } ^ { - k t }$$

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?

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?

6 In a chemical reaction, the mass \(m\) grams of a chemical after \(t\) minutes is modelled by the equation $$m = 20 + 30 \mathrm { e } ^ { - 0.1 t }$$

1. Find the initial mass of the chemical. What is the mass of chemical in the long term?
2. Find the time when the mass is 30 grams.
3. Sketch the graph of \(m\) against \(t\).

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 \(50 ^ { \circ } \mathrm { C }\).

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

1. Find \(a , b\) and \(k\).
2. Find how long it takes for the temperature to reach \(80 ^ { \circ } \mathrm { C }\).

4 The height \(h\) metres of a tree after \(t\) years is modelled by the equation $$h = a - b \mathrm { e } ^ { - k t }$$ 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 \(b\).
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.

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 \(k\).

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. The temperature, \(\theta ^ { \circ } \mathrm { C }\), of a cup of tea \(t\) minutes after it was placed on a table in a room, is modelled by the equation

$$\theta = 18 + 65 \mathrm { e } ^ { - \frac { t } { 8 } } \quad t \geqslant 0$$ Find, according to the model,

1. the temperature of the cup of tea when it was placed on the table,
2. the value of \(t\), to one decimal place, when the temperature of the cup of tea was \(35 ^ { \circ } \mathrm { C }\).
3. Explain why, according to this model, the temperature of the cup of tea could not fall to \(15 ^ { \circ } \mathrm { C }\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{bcbd842f-b2e2-4587-ab4c-15a57a449e5d-16_675_951_973_573} \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{figure} The temperature, \(\mu ^ { \circ } \mathrm { C }\), of a second cup of tea \(t\) minutes after it was placed on a table in a different room, is modelled by the equation $$\mu = A + B \mathrm { e } ^ { - \frac { t } { 8 } } \quad t \geqslant 0$$ where \(A\) and \(B\) are constants.
   Figure 2 shows a sketch of \(\mu\) against \(t\) with two data points that lie on the curve.
   The line \(l\), also shown on Figure 2, is the asymptote to the curve.
   Using the equation of this model and the information given in Figure 2
4. find an equation for the asymptote \(l\).

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

\section*{Solutions relying entirely on calculator technology are not acceptable.} The air pressure, \(P \mathrm {~kg} / \mathrm { cm } ^ { 2 }\), inside a car tyre, \(t\) minutes from the instant when the tyre developed a puncture is given by the equation $$P = k + 1.4 \mathrm { e } ^ { - 0.5 t } \quad t \in \mathbb { R } \quad t \geqslant 0$$ where \(k\) is a constant.
Given that the initial air pressure inside the tyre was \(2.2 \mathrm {~kg} / \mathrm { cm } ^ { 2 }\)

1. state the value of \(k\). From the instant when the tyre developed the puncture,
2. find the time taken for the air pressure to fall to \(1 \mathrm {~kg} / \mathrm { cm } ^ { 2 }\) Give your answer in minutes to one decimal place.
3. Find the rate at which the air pressure in the tyre is decreasing exactly 2 minutes from the instant when the tyre developed the puncture.
   Give your answer in \(\mathrm { kg } / \mathrm { cm } ^ { 2 }\) per minute to 3 significant figures.

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.

9. A cup of tea is cooling down in a room. The temperature of tea, \(\theta ^ { \circ } \mathrm { C }\), at time \(t\) minutes after the tea is made, is modelled by the equation $$\theta = A + 70 e ^ { - 0.025 t }$$ where \(A\) is a positive constant.
Given that the initial temperature of the tea is \(85 ^ { \circ } \mathrm { C }\) a. find the value of \(A\).
b. Find the temperature of the tea 20 minutes after it is made.
c. Find how long it will take the tea to cool down to \(43 ^ { \circ } \mathrm { C }\).
(4)

1. Coffee is poured into a cup.

The temperature of the coffee, \(H ^ { \circ } \mathrm { C } , t\) minutes after being poured into the cup is modelled by the equation $$H = A \mathrm { e } ^ { - B t } + 30$$ where \(A\) and \(B\) are constants.
Initially, the temperature of the coffee was \(85 ^ { \circ } \mathrm { C }\).

1. State the value of \(A\). Initially, the coffee was cooling at a rate of \(7.5 ^ { \circ } \mathrm { C }\) per minute.
2. Find a complete equation linking \(H\) and \(t\), giving the value of \(B\) to 3 decimal places.

1. A quantity of ethanol was heated until it reached boiling point.

The temperature of the ethanol, \(\theta ^ { \circ } \mathrm { C }\), at time \(t\) seconds after heating began, is modelled by the equation $$\theta = A - B \mathrm { e } ^ { - 0.07 t }$$ where \(A\) and \(B\) are positive constants.
Given that

• the initial temperature of the ethanol was \(18 ^ { \circ } \mathrm { C }\)
• after 10 seconds the temperature of the ethanol was \(44 ^ { \circ } \mathrm { C }\)
  1. find a complete equation for the model, giving the values of \(A\) and \(B\) to 3 significant figures.

Ethanol has a boiling point of approximately \(78 ^ { \circ } \mathrm { C }\)

Use this information to evaluate the model.

1. A cup of hot tea was placed on a table. At time \(t\) minutes after the cup was placed on the table, the temperature of the tea in the cup, \(\theta ^ { \circ } \mathrm { C }\), is modelled by the equation

$$\theta = 25 + A \mathrm { e } ^ { - 0.03 t }$$ where \(A\) is a constant. The temperature of the tea was \(75 ^ { \circ } \mathrm { C }\) when the cup was placed on the table.

1. Find a complete equation for the model.
2. Use the model to find the time taken for the tea to cool from \(75 ^ { \circ } \mathrm { C }\) to \(60 ^ { \circ } \mathrm { C }\), giving your answer in minutes to one decimal place. Two hours after the cup was placed on the table, the temperature of the tea was measured as \(20.3 ^ { \circ } \mathrm { C }\). Using this information,
3. evaluate the model, explaining your reasoning.