9 In 2012 Adam bought a second hand car for \(\pounds 8500\). Each year Adam has his car valued. He believes that there is a non-linear relationship between \(t\), the time in years since he bought the car, and \(V\), the value of the car in pounds. Fig. 9.1 shows successive values of \(V\) and \(\log _ { 10 } V\).
\begin{table}[h]
| \(t\) | 0 | 1 | 2 | 3 | 4 |
| \(V\) | 8500 | 6970 | 5720 | 4690 | 3840 |
| \(\log _ { 10 } V\) | 3.93 | 3.84 | 3.76 | 3.67 | 3.58 |
\captionsetup{labelformat=empty}
\caption{Fig. 9.1}
\end{table}
Adam uses a spreadsheet to plot the points ( \(t , \log _ { 10 } V\) ) shown in Fig. 9.1, and then generates a line of best fit for these points. The line passes through the points \(( 0,3.93 )\) and \(( 4,3.58 )\). A copy of his graph is shown in Fig. 9.2.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{11e5167f-9f95-4494-9b66-b59fdce8b1ef-6_776_682_1886_246}
\captionsetup{labelformat=empty}
\caption{Fig. 9.2}
\end{figure}
- Find an expression for \(\log _ { 10 } V\) in terms of \(t\).
- Find a model for \(V\) in the form \(V = A \times b ^ { t }\), where \(A\) and \(b\) are constants to be determined. Give the values of \(A\) and \(b\) correct to 2 significant figures.
In 2017 Adam's car was valued at \(\pounds 3150\).
- Determine whether the model is a good fit for this data.
A company called Webuyoldcars pays \(\pounds 500\) for any second hand car. Adam decides that he will sell his car to this company when the annual valuation of his car is less than \(\pounds 500\).
- According to the model, after how many years will Adam sell his car to Webuyoldcars?