3. A manager wishes to purchase seats for a new cinema. He wishes to buy three types of seat; standard, deluxe and majestic. Let the number of standard, deluxe and majestic seats to be bought be \(x , y\) and \(z\) respectively.
He decides that the total number of deluxe and majestic seats should be at most half of the number of standard seats.
The number of deluxe seats should be at least \(10 \%\) and at most \(20 \%\) of the total number of seats.
The number of majestic seats should be at least half of the number of deluxe seats.
The total number of seats should be at least 250 .
Standard, deluxe and majestic seats each cost \(\pounds 20 , \pounds 26\) and \(\pounds 36\), respectively. The manager wishes to minimize the total cost, \(\pounds C\), of the seats.
Formulate this situation as a linear programming problem, simplifying your inequalities so that all the coefficients are integers.
(9)
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The arcs in Fig. 2 represent roads in a town. The weight on each arc gives the time, in minutes, taken to drive along that road. The times taken to drive along \(A B\) and \(D E\) vary depending upon the time of day.
A police officer wishes to drive along each road at least once, starting and finishing at \(A\). The journey is to be completed in the least time.
- Briefly explain how you know that a route between \(B\) and \(E\) will have to be repeated.
- List the possible routes between \(B\) and \(E\). State how long each would take, in terms of \(x\) where appropriate.
- Find the range of values that \(x\) must satisfy so that \(D E\) would be one of the repeated arcs.
Given that \(x = 7\),
- find the total time needed for the police officer to carry out this journey.