9.

Two models are proposed for the value of a car.
\begin{enumerate}[label=(\alph*)]
\item The first model suggests that the value of the car, $V$ pounds, is given by $V = 18000 - 6000 \sqrt { t }$, where $t$ is the time in years after the car was first purchased.\\
(a) (i) State the value of the car when it was first purchased.\\
(a) (ii) Find $V$ and $\frac { \mathrm { d } V } { \mathrm {~d} t }$ when $t = 4$\\
(a) (iii) Interpret your answers to (a)(ii) in the context of the model.
\item The second model that is proposed suggests that the value of the car, $V$ pounds, is given by $V = a b ^ { - t }$, where $t$ is the time in years after the car was first purchased.

When $t = 0$, both models give the same value for $V$.\\
When $t = 4$, both models give the same value for $V$.

Find the value of $a$ and the value of $b$.\\[0pt]
[3 marks]
\item Explain, with a reason, which model is likely to be the better model over time.
\end{enumerate}

\hfill \mbox{\textit{SPS SPS SM Pure 2023 Q9 [11]}}