Exponential model with shifted asymptote

7. \begin{figure}[h] \end{figure} Figure 1 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 A botanist is investigating the rate of growth of a certain species of toadstool. She observes that a particular toadstool of this type has a height of 57 millimetres at a time 12 hours after it begins to grow. She proposes the model \(h = A \left( 1 - \mathrm { e } ^ { - \frac { 1 } { 4 } t } \right)\), where \(A\) is a constant, for the height \(h\) millimetres of the toadstool, \(t\) hours after it begins to grow.

1. Use this model to:
   1. find the height of the toadstool when \(t = 0\);
   2. show that \(A = 60\), correct to two significant figures.
2. Use the model \(h = 60 \left( 1 - \mathrm { e } ^ { - \frac { 1 } { 4 } t } \right)\) to:
   1. show that the time \(T\) hours for the toadstool to grow to a height of 48 millimetres is given by $$T = a \ln b$$ where \(a\) and \(b\) are integers;
   2. show that \(\frac { \mathrm { d } h } { \mathrm {~d} t } = 15 - \frac { h } { 4 }\);
   3. find the height of the toadstool when it is growing at a rate of 13 millimetres per hour.
      (1 mark)

4 A biologist is researching the growth of a certain species of hamster. She proposes that the length, \(x \mathrm {~cm}\), of a hamster \(t\) days after its birth is given by $$x = 15 - 12 \mathrm { e } ^ { - \frac { t } { 14 } }$$

1. Use this model to find:
   1. the length of a hamster when it is born;
   2. the length of a hamster after 14 days, giving your answer to three significant figures.
2. 1. Show that the time for a hamster to grow to 10 cm in length is given by \(t = 14 \ln \left( \frac { a } { b } \right)\), where \(a\) and \(b\) are integers.
   2. Find this time to the nearest day.
3. 1. Show that $$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { 1 } { 14 } ( 15 - x )$$
   2. Find the rate of growth of the hamster, in cm per day, when its length is 8 cm .
      (1 mark)

10 A bottle of water has a temperature of \(6 ^ { \circ } \mathrm { C }\) when it is removed from a refrigerator. It is placed in a room where the temperature is \(20 ^ { \circ } \mathrm { C }\) 10 minutes later, the temperature of the water is \(12 ^ { \circ } \mathrm { C }\) The temperature of the water, \(T ^ { \circ } \mathrm { C }\), at time \(t\) minutes after it is removed from the refrigerator, may be modelled by the equation $$T = 20 - a \mathrm { e } ^ { - k t }$$ 10

1. Find the value of \(a\). 10
2. Calculate the value of \(k\), giving your answer to two significant figures.
   10
3. Using this model, estimate how long it takes the water to reach a temperature of \(18 ^ { \circ } \mathrm { C }\) after it is taken out of the refrigerator. \(18 ^ { \circ } \mathrm { C }\) after it is taken out of the refrigerator. 10
4. Explain why the model may not be appropriate to predict the temperature of the water three hours after it is taken out of the refrigerator.