Moderate -0.5 This is a straightforward linear programming formulation question requiring translation of verbal constraints into algebraic inequalities and an objective function. While it involves multiple constraints (4-5 inequalities) and requires careful algebraic manipulation to clear fractions, it follows a standard D1 template with no novel problem-solving insight needed. The algebraic simplification is routine (multiplying through by denominators). Slightly easier than average A-level due to being a pure formulation task with no solving or interpretation required.
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 £20, £26 and £36, respectively.
The manager wishes to minimize the total cost, £\(C\), of the seats.
Formulate this situation as a linear programming problem, simplifying your inequalities so that all the coefficients are integers.
[9]
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 £20, £26 and £36, respectively.
The manager wishes to minimize the total cost, £$C$, of the seats.
Formulate this situation as a linear programming problem, simplifying your inequalities so that all the coefficients are integers.
[9]
\hfill \mbox{\textit{Edexcel D1 2003 Q3 [9]}}