2 A car tyre has a slow puncture. Initially the tyre is inflated to a pressure of 34.5 psi . The pressure is checked after 3 days and then again after 5 days. The time \(t\) in days and the pressure, \(P\) psi, are shown in the table below. You are given that the pressure in a car tyre is measured in pounds per square inch (psi).
| \(t\) | 0 | 3 | 5 |
| \(P\) | 34.5 | 29.4 | 27.0 |
The owner of the car believes the relationship between \(P\) and \(t\) may be modelled by a polynomial.
- Explain why it is not possible to use Newton's forward difference interpolation method for these data.
- Use Lagrange's form of the interpolating polynomial to find an interpolating polynomial of degree 2 for these data.
The car owner uses the polynomial found in part (b) to model the relationship between \(P\) and \(t\).
Subsequently it is found that when \(t = 6 , P = 26.0\) and when \(t = 10 , P = 24.4\). - Determine whether the owner's model is a good fit for these data.
- Explain why the model would not be suitable in the long term.