3 A publisher has to choose the price at which to sell a certain new book. The total profit, $\pounds t$, that the publisher will make depends on the price, $\pounds p$. He decides to use a model that includes the following assumptions.

\begin{itemize}
  \item If the price is low, many copies will be sold, but the profit on each copy sold will be small, and the total profit will be small.
  \item If the price is high, the profit on each copy sold will be high, but few copies will be sold, and the total profit will be small.
\end{itemize}

The graphs below show two possible models.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{f2f45d6c-cfdc-455b-ab08-597b06a69f36-05_346_465_1027_374}
\captionsetup{labelformat=empty}
\caption{Model A}
\end{center}
\end{figure}

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{f2f45d6c-cfdc-455b-ab08-597b06a69f36-05_348_476_1025_1080}
\captionsetup{labelformat=empty}
\caption{Model B}
\end{center}
\end{figure}
\begin{enumerate}[label=(\alph*)]
\item Explain how model A is inconsistent with one of the assumptions given above.
\item Given that the equation of the curve in model B is quadratic, show that this equation is of the form $t = k \left( 12 p - p ^ { 2 } \right)$, and find the value of the constant $k$.
\item The publisher needs to make a total profit of at least $\pounds 6400$. Use the equation found in part (b) to find the range of values within which model B suggests that the price of the book must lie.
\item Comment briefly on how realistic model B may be in the following cases.

\begin{itemize}
  \item $p = 0$
  \item $p = 12.1$
\end{itemize}
\end{enumerate}

\hfill \mbox{\textit{OCR H240/02  Q3 [9]}}