7 Sam decided to go on a high-protein diet. Sam's mass in \(\mathrm { kg } , M\), after \(t\) days of following the diet is recorded in Fig. 7.1.
\begin{table}[h]
| \(t\) | 0 | 10 | 20 | 30 |
| \(M\) | 88.3 | 80.05 | 78.7 | 78.85 |
\captionsetup{labelformat=empty}
\caption{Fig. 7.1}
\end{table}
A difference table for the data is shown in Fig. 7.2.
\begin{table}[h]
| \(t\) | \(M\) | \(\Delta M\) | \(\Delta ^ { 2 } M\) | \(\Delta ^ { 3 } M\) |
| 0 | 88.3 | | | |
| | | | |
| 10 | 80.05 | | | |
| | | | |
| 20 | 78.7 | | | |
| | | | |
| 30 | 78.85 | | | |
\captionsetup{labelformat=empty}
\caption{Fig. 7.2}
\end{table}
- Complete the copy of the difference table in the Printed Answer Booklet.
Sam's doctor uses these data to construct a cubic interpolating polynomial to model Sam's mass at time \(t\) days after starting the diet.
- Find the model in the form \(\mathrm { M } = \mathrm { at } ^ { 3 } + \mathrm { bt } ^ { 2 } + \mathrm { ct } + \mathrm { d }\), where \(a , b , c\) and \(d\) are constants to be determined.
Subsequently it is found that when \(\mathrm { t } = 40 , \mathrm { M } = 78.7\) and when \(\mathrm { t } = 50 , \mathrm { M } = 80.05\).
- Determine whether the model is a good fit for these data.
- By completing the extended copy of Fig. 7.2 in the Printed Answer Booklet, explain why a quartic model may be more appropriate for the data.
- Refine the doctor's model to include a quartic term.
- Explain whether the new model for Sam's mass is likely to be appropriate over a longer period of time.