LINEAR PROGRAMMING REVISION QUESTIONS WITH ANSWERS | FREE

Linear Programming Problems for High School Students

Solve the following problems. Use graphical or algebraic methods where possible. Currency is in Tanzanian Shillings (TZS).

1. Solve: \(2x + 3y \leq 12\), \(x + y \geq 4\), \(x \geq 0\), \(y \geq 0\)

2. Maximize \(Z = 5x + 3y\) subject to: \(x + 2y \leq 10\), \(3x + y \leq 12\), \(x, y \geq 0\)

3. A farmer has 60 hours of labor and 48 units of fertilizer. Maize needs 3 hrs and 4 units, beans need 4 hrs and 2 units. Profit per acre: TZS 15,000 for maize, TZS 10,000 for beans. Maximize profit.

4. Solve: \(3x - 2y \geq 6\), \(x + 4y \leq 20\), \(x, y \geq 0\)

5. Maximize \(P = 8x + 7y\), subject to: \(2x + y \leq 16\), \(x + 2y \leq 14\), \(x, y \geq 0\)

6. A student wants to spend no more than 20 hours on homework and games. Homework takes 2 hours per subject, games 1 hour per session. At least 4 homework sessions. Maximize fun if each game gives 10 points.

7. Minimize \(C = 3x + 4y\), given: \(x + y \geq 8\), \(2x + 3y \geq 18\), \(x, y \geq 0\)

8. Maximize \(Z = 6x + 5y\), \(x + y \leq 10\), \(2x + y \leq 14\), \(x, y \geq 0\)

9. A shop sells pens and rulers. Pen profit: TZS 1000, ruler profit: TZS 500. Max labor: 20 hrs. Pen needs 2 hrs, ruler 1 hr. Max materials: 30 units. Pen: 3 units, ruler: 2. Maximize profit.

10. Solve: \(x + 2y \geq 10\), \(2x + y \leq 16\), \(x, y \geq 0\)

11. A carpenter makes tables and chairs. Table requires 4 hrs woodwork, 2 hrs polishing. Chair requires 2 hrs woodwork, 1 hr polishing. Max 100 hrs woodwork, 40 hrs polishing. Profit: TZS 40,000 per table, TZS 25,000 per chair. Maximize profit.

12. A bakery makes cakes and bread. Cake: 3 units flour, 2 sugar. Bread: 2 flour, 1 sugar. Available: 90 flour, 50 sugar. Profit: Cake TZS 5,000, Bread TZS 3,000. Maximize profit.

13. A worker makes shirts and trousers. Shirt: 1 hr cutting, 2 hrs sewing. Trouser: 2 hrs cutting, 3 hrs sewing. Max 40 hrs cutting, 60 hrs sewing. Maximize profit: Shirt TZS 15,000, Trouser TZS 20,000.

14. Maximize \(P = 7x + 9y\), \(x + y \leq 10\), \(x \leq 7\), \(y \leq 8\), \(x, y \geq 0\)

15. Minimize \(C = 2x + 5y\), subject to: \(x + y \geq 6\), \(2x + y \geq 8\), \(x, y \geq 0\)

16. A contractor uses trucks and trailers. Truck carries 10 tons, trailer 6 tons. 60 tons required. Trucks cost TZS 20,000/day, trailers TZS 15,000/day. Minimize transport cost.

17. Maximize \(Z = 4x + 6y\) subject to: \(x + y \leq 9\), \(x \leq 5\), \(y \leq 6\), \(x, y \geq 0\)

18. A gardener grows roses and tulips. Rose: 4 sq.m, Tulip: 2 sq.m. Max 100 sq.m. Budget: Rose TZS 2000, Tulip TZS 1000. Budget limit TZS 40,000. Maximize flowers planted.

19. A supplier delivers by van and lorry. Van: 3 tons, Lorry: 7 tons. Need 84 tons. Van cost TZS 50,000, Lorry TZS 100,000. Minimize cost.

20. Maximize \(P = 5x + 4y\), subject to: \(x + y \leq 10\), \(x + 2y \leq 12\), \(x, y \geq 0\)

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Answers

1. Feasible region vertices: (0,4), (3,2), (6,0)
2. Max at (2,4), \(Z = 22\)
3. Maize: 8 acres, Beans: 6 acres, Profit = TZS 210,000
4. Region includes (0,0), (0,5), (4,4), (6,0)
5. Max at (6,2), \(P = 62\)
6. Games = 6, Homework = 4, Fun points = 60
7. Min at (2,4), \(C = 22\)
8. Max at (4,2), \(Z = 34\)
9. Max profit = TZS 13,000 at 5 pens and 5 rulers
10. Region bounded by intersection points (0,10), (2,6), (8,0)
11. Tables = 10, Chairs = 20, Profit = TZS 1,000,000
12. Cakes = 10, Bread = 15, Profit = TZS 95,000
13. Shirts = 10, Trousers = 10, Profit = TZS 350,000
14. Max at (2,8), \(P = 86\)
15. Min at (2,4), \(C = 24\)
16. Trucks = 3, Trailers = 5, Cost = TZS 195,000
17. Max at (3,6), \(Z = 54\)
18. Roses = 10, Tulips = 20, Max flowers = 30
19. Vans = 4, Lorries = 8, Cost = TZS 1,000,000
20. Max at (4,4), \(P = 36\)