7.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{5fa867a2-0a3d-4f0b-9f9c-15584f2be5c0-08_1372_1769_278_189}
\captionsetup{labelformat=empty}
\caption{Figure 6}
\end{center}
\end{figure}

Keith organises two types of children's activity, 'Sports Mad' and 'Circus Fun'. He needs to determine the number of times each type of activity is to be offered.

Let $x$ be the number of times he offers the 'Sports Mad' activity. Let $y$ be the number of times he offers the 'Circus Fun' activity.

Two constraints are

$$\text { and } \quad \begin{aligned}
& x \leqslant 15 \\
& y > 6
\end{aligned}$$

These constraints are shown on the graph in Figure 6, where the rejected regions are shaded out.
\begin{enumerate}[label=(\alph*)]
\item Explain why $y = 6$ is shown as a dotted line.

Two further constraints are

$$\begin{aligned}
& 3 x \geqslant 2 y \\
\text { and } \quad 5 x + 4 y & \geqslant 80
\end{aligned}$$
\item Add two lines and shading to Diagram 1 in the answer book to represent these inequalities. Hence determine the feasible region and label it R .

Each 'Sports Mad' activity costs $\pounds 500$.\\
Each 'Circus Fun' activity costs $\pounds 800$.\\
Keith wishes to minimise the total cost.
\item Write down the objective function, C , in terms of $x$ and $y$.
\item Use your graph to determine the number of times each type of activity should be offered and the total cost. You must show sufficient working to make your method clear.
\end{enumerate}

\hfill \mbox{\textit{Edexcel D1 2013 Q7 [11]}}