Critique single model appropriateness

1. In a simple model, the value, \(\pounds V\), of a car depends on its age, \(t\), in years.

The following information is available for \(\operatorname { car } A\)

• its value when new is \(\pounds 20000\)
• its value after one year is \(\pounds 16000\)
  1. Use an exponential model to form, for car \(A\), a possible equation linking \(V\) with \(t\).

The value of car \(A\) is monitored over a 10-year period.
Its value after 10 years is \(\pounds 2000\)

Evaluate the reliability of your model in light of this information. The following information is available for car \(B\)

• it has the same value, when new, as car \(A\)
• its value depreciates more slowly than that of \(\operatorname { car } A\)
• Explain how you would adapt the equation found in (a) so that it could be used to model the value of car \(B\).

12 In an experiment 500 fruit flies were released into a controlled environment. After 10 days there were 650 fruit flies present. Munirah believes that \(N\), the number of fruit flies present at time \(t\) days after the fruit flies are released, will increase at the rate of \(4.4 \%\) per day. She proposes that the situation is modelled by the formula \(N = A k ^ { t }\).

1. Write down the values of \(A\) and \(k\).
2. Determine whether the model is consistent with the value of \(N\) at \(t = 10\).
3. What does the model suggest about the number of fruit flies in the long run? Subsequently it is found that for large values of \(t\) the number of fruit flies in the controlled environment oscillates about 750 . It is also found that as \(t\) increases the oscillations decrease in magnitude. Munirah proposes a second model in the light of this new information. $$N = 750 - 250 \times \mathrm { e } ^ { - 0.092 t } .$$
4. Identify three ways in which this second model is consistent with the known data.
5. (A) Identify one feature which is not accounted for by the second model.
   (B) Give an example of a mathematical function which needs to be incorporated in the model to account for this feature. \section*{END OF QUESTION PAPER}

8 In an experiment, the temperature of a hot liquid is measured every minute.
The difference between the temperature of the hot liquid and room temperature is \(D ^ { \circ } \mathrm { C }\) at time \(t\) minutes. Fig. 8 shows the experimental data. \begin{figure}[h] \end{figure} It is thought that the model \(D = 70 \mathrm { e } ^ { - 0.03 t }\) might fit the data.

1. Write down the derivative of \(\mathrm { e } ^ { - 0.03 t }\).
2. Explain how you know that \(70 \mathrm { e } ^ { - 0.03 t }\) is a decreasing function of \(t\).
3. Calculate the value of \(70 \mathrm { e } ^ { - 0.03 t }\) when
   1. \(\quad t = 0\),
   2. \(t = 20\).
4. Using your answers to parts (b) and (c), discuss how well the model \(D = 70 \mathrm { e } ^ { - 0.03 t }\) fits the data.

16 In the first year of a course, an A-level student, Aaishah, has a mathematics test each week. The night before each test she revises for \(t\) hours. Over the course of the year she realises that her percentage mark for a test, \(p\), may be modelled by the following formula, where \(A , B\) and \(C\) are constants. $$p = A - B ( t - C ) ^ { 2 }$$

• Aaishah finds that, however much she revises, her maximum mark is achieved when she does 2 hours revision. This maximum mark is 62 .
• Aaishah had a mark of 22 when she didn't spend any time revising.
  1. Find the values of \(A , B\) and \(C\).
  2. According to the model, if Aaishah revises for 45 minutes on the night before the test, what mark will she achieve?
  3. What is the maximum amount of time that Aaishah could have spent revising for the model to work?

In an attempt to improve her marks Aaishah now works through problems for a total of \(t\) hours over the three nights before the test. After taking a number of tests, she proposes the following new formula for \(p\). $$p = 22 + 68 \left( 1 - \mathrm { e } ^ { - 0.8 t } \right)$$ For the next three tests she recorded the data in Fig. 16. \begin{table}[h] \end{table}

Verify that the data is consistent with the new formula.

Aaishah's tutor advises her to spend a minimum of twelve hours working through problems in future. Determine whether or not this is good advice.

9 A small country started using solar panels to produce electrical energy in the year 2000. Electricity production is measured in megawatt hours (MWh). For the period from 2000 to 2009, the annual electrical energy produced using solar panels can be modelled by the equation \(\mathrm { P } = 0.3 \mathrm { e } ^ { 0.5 \mathrm { t } }\), where \(P\) is the annual amount of electricity produced in MWh and \(t\) is the time in years after the year 2000.

1. According to this model, find the amount of electricity produced using solar panels in each of the following years.
   1. 2000
   2. 2009
2. Give a reason why the model is unlikely to be suitable for predicting the annual amount of electricity produced using solar panels in the year 2025. An alternative model is suggested; the curve representing this model is shown in Fig. 9. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Fig. 9} \includegraphics[alt={},max width=\textwidth]{20639e13-01cc-4d96-b694-fb3cf1828f4d-08_702_1587_1265_230}
   \end{figure}
3. Explain how the graph shows that the alternative model gives a value for the amount of electricity produced in 2009 that is consistent with the original model.
4. 1. On the axes given in the Printed Answer Booklet, sketch the gradient function of the model shown in Fig. 9.
   2. State approximately the value of \(t\) at the point of inflection in Fig. 9.
   3. Interpret the significance of the point of inflection in the context of the model.
5. State approximately the long term value of the annual amount of electricity produced using solar panels according to the model represented in Fig. 9.

9 An analyst believes that the sales of a particular electronic device are growing exponentially. In 2015 the sales were 3.1 million devices and the rate of increase in the annual sales is 0.8 million devices per year.

1. Find a model to represent the annual sales, defining any variables used.
2. In 2017 the sales were 5.2 million devices. Determine whether this is consistent with the model in part (i).
3. The analyst uses the model in part (i) to predict the sales for 2025. Comment on the reliability of this prediction.

3 The number of members of a social networking site is modelled by \(m = 150 \mathrm { e } ^ { 2 t }\), where \(m\) is the number of members and \(t\) is time in weeks after the launch of the site.

1. State what this model implies about the relationship between \(m\) and the rate of change of \(m\).
2. What is the significance of the integer 150 in the model?
3. Find the week in which the model predicts that the number of members first exceeds 60000 .
4. The social networking site only expects to attract 60000 members. Suggest how the model could be refined to take account of this.

10 On 18 March 2019 there were 12 hours of daylight in Inverness.
On 16 June 2019, 90 days later, there will be 18 hours of daylight in Inverness.
Jude decides to model the number of hours of daylight in Inverness, \(N\), by the formula $$N = A + B \sin t ^ { \circ }$$ where \(t\) is the number of days after 18 March 2019.
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1. 1. State the value that Jude should use for \(A\).
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2. (ii) State the value that Jude should use for \(B\).
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3. (iii) Using Jude's model, calculate the number of hours of daylight in Inverness on 15 May 2019, 58 days after 18 March 2019.
   [0pt] [1 mark]
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4. (iv) Using Jude's model, find how many days during 2019 will have at least 17.4 hours of daylight in Inverness.
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5. (v) Explain why Jude's model will become inaccurate for 2020 and future years.
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6. Anisa decides to model the number of hours of daylight in Inverness with the formula $$N = A + B \sin \left( \frac { 360 } { 365 } t \right) \circ$$ Explain why Anisa's model is better than Jude's model.