Compare or choose between models

4 A species of animal is to be introduced onto a remote island. Their food will consist only of various plants that grow on the island. A zoologist proposes two possible models for estimating the population \(P\) after \(t\) years. The diagrams show these models as they apply to the first 20 years. \includegraphics[max width=\textwidth, alt={}, center]{1a0e0afb-81be-45d1-8c86-f98e508e9a49-05_725_606_406_242} \includegraphics[max width=\textwidth, alt={}, center]{1a0e0afb-81be-45d1-8c86-f98e508e9a49-05_714_593_413_968}

1. Without calculation, describe briefly how the rate of growth of \(P\) will vary for the first 20 years, according to each of these two models. The equation of the curve for model A is \(P = 20 + 1000 \mathrm { e } ^ { - \frac { ( t - 20 ) ^ { 2 } } { 100 } }\).
   The equation of the curve for model B is \(P = 20 + 1000 \left( 1 - \mathrm { e } ^ { - \frac { t } { 5 } } \right)\).
2. Describe the behaviour of \(P\) that is predicted for \(t > 20\)
   1. using model A,
   2. using model B . There is only a limited amount of food available on the island, and the zoologist assumes that the size of the population depends on the amount of food available and on no other external factors.
3. State what is suggested about the long-term food supply by
   1. model A,
   2. model B.

6. A company plans to extract oil from an oil field. The daily volume of oil \(V\), measured in barrels that the company will extract from this oil field depends upon the time, \(t\) years, after the start of drilling. The company decides to use a model to estimate the daily volume of oil that will be extracted. The model includes the following assumptions:

• The initial daily volume of oil extracted from the oil field will be 16000 barrels.
• The daily volume of oil that will be extracted exactly 4 years after the start of drilling will be 9000 barrels.
• The daily volume of oil extracted will decrease over time.

The diagram below shows the graphs of two possible models. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure}

1. 1. Use model \(A\) to estimate the daily volume of oil that will be extracted exactly 3 years after the start of drilling.
   2. Write down a limitation of using model \(A\).
2. 1. Using an exponential model and the information given in the question, find a possible equation for model \(B\).
   2. Using your answer to (b)(i) estimate the daily volume of oil that will be extracted exactly 3 years after the start of drilling.

11 On the day that a new consumer product went on sale (day zero), a call centre received 1 call about it. On the 2nd day after day zero the call centre received 3 calls, and on the 10th day after day zero there were 200 calls. Two models were proposed to model \(N\), the number of calls received \(t\) days after day zero.
Model 1 is a linear model \(\mathrm { N } = \mathrm { mt } + \mathrm { c }\).

1. Determine the values of \(m\) and \(c\) which best model the data for 2 days and 10 days after day zero.
2. State the rate of increase in calls according to model 1.
3. Explain why this model is not suitable when \(t = 1\). Model 2 is an exponential model \(\mathbf { N } = e ^ { 0.53 t }\).
4. Verify that this is a good model for the number of calls when \(t = 2\) and \(t = 10\).
5. Determine the rate of increase in calls when \(t = 10\) according to model 2 .

10 Zac is measuring the growth of a culture of bacteria in a laboratory. The initial area of the culture is \(8 \mathrm {~cm} ^ { 2 }\). The area one day later is \(8.8 \mathrm {~cm} ^ { 2 }\). At first, Zac uses a model of the form \(\mathrm { A } = \mathrm { a } + \mathrm { bt }\), where \(A \mathrm {~cm} ^ { 2 }\) is the area \(t\) days after he begins measuring and \(a\) and \(b\) are constants.

1. Find the values of \(a\) and \(b\) that best model the initial area and the area one day later.
2. Calculate the value of \(t\) for which the model predicts an area of \(15 \mathrm {~cm} ^ { 2 }\).
3. Zac notices the area covered by the culture increases by \(10 \%\) each day. Explain why this model may not be suitable after the first day. Zac decides to use a different model for \(A\). His new model is \(\mathrm { A } = \mathrm { Pe } ^ { \mathrm { kt } }\), where \(P\) and \(k\) are constants.
4. Find the values of \(P\) and \(k\) that best model the initial area and the area one day later.
5. Calculate the value of \(t\) for which the area reaches \(15 \mathrm {~cm} ^ { 2 }\) according to this model.
6. Explain why this model may not be suitable for large values of \(t\).