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]
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
$t$ & 0 & 1 & 2 & 3 & 4 \\
\hline
$V$ & 8500 & 6970 & 5720 & 4690 & 3840 \\
\hline
$\log _ { 10 } V$ & 3.93 & 3.84 & 3.76 & 3.67 & 3.58 \\
\hline
\end{tabular}
\captionsetup{labelformat=empty}
\caption{Fig. 9.1}
\end{center}
\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]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{11e5167f-9f95-4494-9b66-b59fdce8b1ef-6_776_682_1886_246}
\captionsetup{labelformat=empty}
\caption{Fig. 9.2}
\end{center}
\end{figure}
\begin{enumerate}[label=(\alph*)]
\item Find an expression for $\log _ { 10 } V$ in terms of $t$.
\item 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 $\pounds 3150$.
\item 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 $\pounds 500$.
\item According to the model, after how many years will Adam sell his car to Webuyoldcars?
\end{enumerate}

\hfill \mbox{\textit{OCR MEI AS Paper 2 2019 Q9 [10]}}