Formulation with percentage constraints

3. Donald plans to bake and sell cakes. The three types of cake that he can bake are brownies, flapjacks and muffins. Donald decides to bake 48 brownies and muffins in total.
Donald decides to bake at least 5 brownies for every 3 flapjacks.
At most \(40 \%\) of the cakes will be muffins.
Donald has enough ingredients to bake 60 brownies or 45 flapjacks or 35 muffins.
Donald plans to sell each brownie for \(\pounds 1.50\), each flapjack for \(\pounds 1\) and each muffin for \(\pounds 1.25\) He wants to maximise the total income from selling the cakes. Let \(x\) represent the number of brownies, let \(y\) represent the number of flapjacks and let \(z\) represent the number of muffins that Donald will bake. Formulate this as a linear programming problem in \(x\) and \(y\) only, stating the objective function and listing the constraints as simplified inequalities with integer coefficients. You should not attempt to solve the problem.

1. Jonathan makes two types of information pack for an event, Standard and Value.

Each Standard pack contains 25 posters and 500 flyers.
Each Value pack contains 15 posters and 800 flyers.
He must use at least 150000 flyers.
Between \(35 \%\) and \(65 \%\) of the packs must be Standard packs.
Posters cost 20p each and flyers cost 4p each.
Jonathan wishes to minimise his costs.
Let x and y represent the number of Standard packs and Value packs produced respectively.
Formulate this as a linear programming problem, stating the objective and listing the constraints as simplified inequalities with integer coefficients. You should not attempt to solve the problem. \section*{(Total for Question 5 is 5 marks)} TOTAL IS 40 MARKS

9 A factory makes three different types of widget: plain, bland and ordinary. Each widget is made using three different machines: \(A , B\) and \(C\). Each plain widget needs 5 minutes on machine \(A , 12\) minutes on machine \(B\) and 24 minutes on machine \(C\). Each bland widget needs 4 minutes on machine \(A , 8\) minutes on machine \(B\) and 12 minutes on machine \(C\). Each ordinary widget needs 3 minutes on machine \(A\), 10 minutes on machine \(B\) and 18 minutes on machine \(C\). Machine \(A\) is available for 3 hours a day, machine \(B\) for 4 hours a day and machine \(C\) for 9 hours a day. The factory must make:
more plain widgets than bland widgets;
more bland widgets than ordinary widgets.
At least \(40 \%\) of the total production must be plain widgets.
Each day, the factory makes \(x\) plain, \(y\) bland and \(z\) ordinary widgets.
Formulate the above situation as 6 inequalities, in addition to \(x \geqslant 0 , y \geqslant 0\) and \(z \geqslant 0\), writing your answers with simplified integer coefficients.
(8 marks)

8 Each day, a factory makes three types of hinge: basic, standard and luxury. The hinges produced need three different components: type \(A\), type \(B\) and type \(C\). Basic hinges need 2 components of type \(A , 3\) components of type \(B\) and 1 component of type \(C\). Standard hinges need 4 components of type \(A , 2\) components of type \(B\) and 3 components of type \(C\). Luxury hinges need 3 components of type \(A\), 4 components of type \(B\) and 5 components of type \(C\). Each day, there are 360 components of type \(A\) available, 270 of type \(B\) and 450 of type \(C\). Each day, the factory must use at least 720 components in total.
Each day, the factory must use at least \(40 \%\) of the total components as type \(A\).
Each day, the factory makes \(x\) basic hinges, \(y\) standard hinges and \(z\) luxury hinges.
In addition to \(x \geqslant 0 , y \geqslant 0 , z \geqslant 0\), find five inequalities, each involving \(x , y\) and \(z\), which must be satisfied. Simplify each inequality where possible.

8 A factory packs three different kinds of novelty box: red, blue and green. Each box contains three different types of toy: \(\mathrm { A } , \mathrm { B }\) and C . Each red box has 2 type A toys, 3 type B toys and 4 type C toys.
Each blue box has 3 type A toys, 1 type B toy and 3 type C toys.
Each green box has 4 type A toys, 5 type B toys and 2 type C toys.
Each day, the maximum number of each type of toy available to be packed is 360 type A, 300 type B and 400 type C. Each day, the factory must pack more type A toys than type B toys.
Each day, the total number of type A and type B toys that are packed must together be at least as many as the number of type C toys that are packed. Each day, at least \(40 \%\) of the total toys that are packed must be type C toys.
Each day, the factory packs \(x\) red boxes, \(y\) blue boxes and \(z\) green boxes.
Formulate the above situation as 6 inequalities, in addition to \(x \geqslant 0 , y \geqslant 0\) and \(z \geqslant 0\), simplifying your answers.

7 A factory makes batches of three different types of battery: basic, long-life and super.
Each basic batch needs 4 minutes on machine \(A\), 7 minutes on machine \(B\) and 14 minutes on machine \(C\). Each long-life batch needs 10 minutes on machine \(A\), 14 minutes on machine \(B\) and 21 minutes on machine \(C\). Each super batch needs 10 minutes on machine \(A\), 14 minutes on machine \(B\) and 28 minutes on machine \(C\). Machine \(A\) is available for 4 hours a day, machine \(B\) for 3.5 hours a day and machine \(C\) for 7 hours a day. Each day the factory must make:
more basic batches than the total number of long-life and super batches; at least as many long-life batches as super batches. At least 15\% of the production must be long-life batches.
Each day, the factory makes \(x\) basic, \(y\) long-life and \(z\) super batches.
Formulate the above situation as 6 inequalities, in addition to \(x \geqslant 0 , y \geqslant 0\) and \(z \geqslant 0\), writing your answers with simplified integer coefficients.

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]

The Young Enterprise Company "Decide", is going to produce badges to sell to decision maths students. It will produce two types of badges. Badge 1 reads "I made the decision to do maths" and Badge 2 reads "Maths is the right decision". "Decide" must produce at least 200 badges and has enough material for 500 badges. Market research suggests that the number produced of Badge 1 should be between 20% and 40% of the total number of badges made. The company makes a profit of 30p on each Badge 1 sold and 40p on each Badge 2. It will sell all that it produced, and wishes to maximise its profit. Let \(x\) be the number produced of Badge 1 and \(y\) be the number of Badge 2.

1. Formulate this situation as a linear programming problem, simplifying your inequalities so that all the coefficients are integers. [6]
2. On the grid provided in the answer book, construct and clearly label the feasible region. [5]
3. Using your graph, advise the company on the number of each badge it should produce. State the maximum profit "Decide" will make. [3]

Phil is to buy some squash balls for his club. There are three different types of ball that he can buy: slow, medium and fast. He must buy at least 190 slow balls, at least 50 medium balls and at least 50 fast balls. He must buy at least 300 balls in total. Each slow ball costs £2.50, each medium ball costs £2.00 and each fast ball costs £2.00. He must spend no more than £1000 in total. At least 60% of the balls that he buys must be slow balls. Phil buys \(x\) slow balls, \(y\) medium balls and \(z\) fast balls.

1. Find six inequalities that model Phil's situation. [4 marks]
2. Phil decides to buy the same number of medium balls as fast balls.
   1. Show that the inequalities found in part (a) simplify to give $$x \geq 190, \quad y \geq 50, \quad x + 2y \geq 300, \quad 5x + 8y \leq 2000, \quad y \leq \frac{1}{3}x$$ [2 marks]
   2. Phil sells all the balls that he buys to members of the club. He sells each slow ball for £3.00, each medium ball for £2.25 and each fast ball for £2.25. He wishes to maximise his profit. On Figure 1 on page 14, draw a diagram to enable this problem to be solved graphically, indicating the feasible region and the direction of an objective line. [7 marks]
   3. Find Phil's maximum possible profit and state the number of each type of ball that he must buy to obtain this maximum profit. [4 marks]