7. A leisure company owns boats of each of the following types: 2-person boats which are 4 metres long and weigh 50 kg .
4-person boats which are 3 metres long and weigh 20 kg .
8-person boats which are 14 metres long and weigh 100 kg .
The leisure company is willing to donate boats to a local sports club to accommodate up to 40 people at any one time. However, storage facilities mean that a maximum combined length of the boats must not be more than 75 metres. Also, it must be possible to transport all the boats on a single trailer which has a maximum load capacity of 600 kg . The club intends to hire the boats out to help with the cost of maintaining them. It plans to charge \(\pounds 10 , \pounds 12\) and \(\pounds 8\) per day, for the 2 -, 4 - and 8 -person boats respectively and wishes to maximise its daily revenue ( \(\pounds R\) ). Let \(x , y\) and \(z\) represent the number of 2-, 4- and 8-person boats respectively given to the club.

1. Model this as a linear programming problem simplifying your expressions so that they have integer coefficients.
   (4 marks)
2. Show that the initial tableau, when using the simplex algorithm, can be written as:

   Basic Variable	\(x\)	\(y\)	\(z\)	\(s\)	\(t\)	\(u\)	Value
   \(s\)	1	2	4	1	0	0	20
   \(t\)	4	3	14	0	1	0	75
   \(u\)	5	2	10	0	0	1	60
   \(R\)	\({ } ^ { - } 10\)	\({ } ^ { - } 12\)	\({ } ^ { - } 8\)	0	0	0	0

3. Explain the purpose of the variables \(s\), \(t\) and \(u\).
4. By increasing the value of \(y\) first, work out the next two complete tableaus.
5. Explain how you know that your final tableau gives an optimal solution and state this solution in practical terms. Sheet for answering question 3
   NAME \section*{Please hand this sheet in for marking}
8. 
   \includegraphics[max width=\textwidth, alt={}, center]{6c6b7934-ab46-4a87-8a11-f99bf9a5d743-08_2017_1051_462_244}
   \section*{Please hand this sheet in for marking}
9. \(F \quad \bullet\)
   • \(W\)
     \(G \quad \bullet\)
   • \(S\)
   H •
   • \(C\)
   I •
   • \(O\)
   J •
   • \(D\)
   • Initial matching:
     \(\begin{array} { l l l l } F & \bullet & \bullet & W
     G & \bullet & \bullet & S
     H & \bullet & \bullet & C
     I & \bullet & \bullet & O
     J & \bullet & \bullet & D \end{array}\) \(\_\_\_\_\)
     
   Complete matching:
   F •
   • \(W\)
     \(G \quad \bullet\)
   • \(S\)
     H •
   • \(C\)
     I •
   • \(O\)
     J •
   • \(D\)
   \section*{Sheet for answering question 5} NAME \section*{Please hand this sheet in for marking}
   \includegraphics[max width=\textwidth, alt={}]{6c6b7934-ab46-4a87-8a11-f99bf9a5d743-10_2398_643_248_1224}
   Sheet for answering question 6
   NAME \section*{Please hand this sheet in for marking}
10. \(\_\_\_\_\)
11. \(\_\_\_\_\)
12. \(\_\_\_\_\)
13. 
    \includegraphics[max width=\textwidth, alt={}, center]{6c6b7934-ab46-4a87-8a11-f99bf9a5d743-11_592_1292_1078_312}
    Sheet for answering question 6 (cont.)
    \includegraphics[max width=\textwidth, alt={}, center]{6c6b7934-ab46-4a87-8a11-f99bf9a5d743-12_595_1299_351_312}
    \includegraphics[max width=\textwidth, alt={}, center]{6c6b7934-ab46-4a87-8a11-f99bf9a5d743-12_597_1298_1409_308}