12.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3f6f3f19-a1d0-488b-a1a4-302cc4cf5a1e-34_643_652_210_708}
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\caption{Figure 2}
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The number of subscribers to two different music streaming companies is being monitored.
The number of subscribers, \(N _ { \mathrm { A } }\), in thousands, to company \(\mathbf { A }\) is modelled by the equation
$$N _ { \mathrm { A } } = | t - 3 | + 4 \quad t \geqslant 0$$
where \(t\) is the time in years since monitoring began.
The number of subscribers, \(N _ { \mathrm { B } }\), in thousands, to company B is modelled by the equation
$$N _ { \mathrm { B } } = 8 - | 2 t - 6 | \quad t \geqslant 0$$
where \(t\) is the time in years since monitoring began.
Figure 2 shows a sketch of the graph of \(N _ { \mathrm { A } }\) and the graph of \(N _ { \mathrm { B } }\) over a 5-year period.
Use the equations of the models to answer parts (a), (b), (c) and (d).
- Find the initial difference between the number of subscribers to company \(\mathbf { A }\) and the number of subscribers to company B.
When \(t = T\) company A reduced its subscription prices and the number of subscribers increased.
- Suggest a value for \(T\), giving a reason for your answer.
- Find the range of values of \(t\) for which \(N _ { \mathrm { A } } > N _ { \mathrm { B } }\) giving your answer in set notation.
- State a limitation of the model used for company B.