Linear modelling problems

10 Simplify \(\left( m ^ { 2 } + 1 \right) ^ { 2 } - \left( m ^ { 2 } - 1 \right) ^ { 2 }\), showing your method.
Hence, given the right-angled triangle in Fig. 10, express \(p\) in terms of \(m\), simplifying your answer. \begin{figure}[h] \end{figure}

3 Ayesha, Bob, Chloe and Dave are discussing the relationship between the time, \(t\) hours, they might spend revising for an examination, and the mark, \(m\), they would expect to gain. Each of them draws a graph to model this relationship for himself or herself.
\includegraphics[max width=\textwidth, alt={}, center]{65d9d34c-8c78-45fe-b9f0-dab071ae56bb-04_437_423_1576_187}
\includegraphics[max width=\textwidth, alt={}, center]{65d9d34c-8c78-45fe-b9f0-dab071ae56bb-04_439_426_1576_609}
\includegraphics[max width=\textwidth, alt={}, center]{65d9d34c-8c78-45fe-b9f0-dab071ae56bb-04_439_428_1576_1032}
\includegraphics[max width=\textwidth, alt={}, center]{65d9d34c-8c78-45fe-b9f0-dab071ae56bb-04_437_419_1576_1462}

1. Assuming Ayesha's model is correct, how long would you recommend that she spends revising?
2. State one feature of Dave's model that is likely to be unrealistic.
3. Suggest a reason for the shape of Bob's graph as compared with Ayesha's graph.
4. What does Chloe's model suggest about her attitude to revision?

1. In 1997 the average \(\mathrm { CO } _ { 2 }\) emissions of new cars in the UK was \(190 \mathrm {~g} / \mathrm { km }\).

In 2005 the average \(\mathrm { CO } _ { 2 }\) emissions of new cars in the UK had fallen to \(169 \mathrm {~g} / \mathrm { km }\).
Given \(\mathrm { Ag } / \mathrm { km }\) is the average \(\mathrm { CO } _ { 2 }\) emissions of new cars in the UK \(n\) years after 1997 and using a linear model,

1. form an equation linking \(A\) with \(n\). In 2016 the average \(\mathrm { CO } _ { 2 }\) emissions of new cars in the UK was \(120 \mathrm {~g} / \mathrm { km }\).
2. Comment on the suitability of your model in light of this information.

1. The distance a particular car can travel in a journey starting with a full tank of fuel was investigated.

• From a full tank of fuel, 40 litres remained in the car's fuel tank after the car had travelled 80 km
• From a full tank of fuel, 25 litres remained in the car's fuel tank after the car had travelled 200 km

Using a linear model, with \(V\) litres being the volume of fuel remaining in the car's fuel tank and \(d \mathrm {~km}\) being the distance the car had travelled,

1. find an equation linking \(V\) with \(d\). Given that, on a particular journey
   • the fuel tank of the car was initially full
   • the car continued until it ran out of fuel
     find, according to the model,
   • 1. the initial volume of fuel that was in the fuel tank of the car,
     2. the distance that the car travelled on this journey.
   In fact the car travelled 320 km on this journey.
2. Evaluate the model in light of this information.

1. A tank, which contained water, started to leak from a hole in its base.

The volume of water in the tank 24 minutes after the leak started was \(4 \mathrm {~m} ^ { 3 }\) The volume of water in the tank 60 minutes after the leak started was \(2.8 \mathrm {~m} ^ { 3 }\) The volume of water, \(V \mathrm {~m} ^ { 3 }\), in the tank \(t\) minutes after the leak started, can be described by a linear model between \(V\) and \(t\).

1. Find an equation linking \(V\) with \(t\). Use this model to find
2. 1. the initial volume of water in the tank,
   2. the time taken for the tank to empty.
3. Suggest a reason why this linear model may not be suitable.

11 David puts a block of ice into a cool-box. He wishes to model the mass \(m \mathrm {~kg}\) of the remaining block of ice at time \(t\) hours later. He finds that when \(t = 5 , m = 2.1\), and when \(t = 50 , m = 0.21\).

1. David at first guesses that the mass may be inversely proportional to time. Show that this model fits his measurements.
2. Explain why this model
   1. is not suitable for small values of \(t\),
   2. cannot be used to find the time for the block to melt completely. David instead proposes a linear model \(m = a t + b\), where \(a\) and \(b\) are constants.
3. Find the values of the constants for which the model fits the mass of the block when \(t = 5\) and \(t = 50\).
4. Interpret these values of \(a\) and \(b\).
5. Find the time according to this model for the block of ice to melt completely.

12 Explain why the smaller regular hexagon in Fig. C1 has perimeter 6.

9 A craft artist is producing items and selling them in a local market. The selling price, \(\pounds P\), of an item is inversely proportional to the number of items produced, \(n\) 9

1. When \(n = 10 , P = 24\)
   Show that \(P = \frac { 240 } { n }\) 9
2. The production cost, \(\pounds C\), of an item is inversely proportional to the square of the number of items produced, \(n\) When \(n = 10 , C = 60\) Find the set of values of \(n\) for which \(P > C\)
   9
3. Explain the significance to the craft artist of the range of values found in part (b).
   [0pt] [1 mark]