1. NCEA Questions

2. Question 3 2004 Andy’s confectionery shop sells sweets. Bags of wine gums and jaffas are put together in two different combinations. Small bags are made up of 250 grams of wine gums and 250 grams of jaffas. Big bags are made up of 250 grams of wine gums and 750 grams of jaffas. Andy has 180 kilograms of wine gums and 240 kilograms of jaffas in stock. Let the number of small bags Andy sells be s and the number of big bags he sells be b.

3. A linear programming problem for this situation has the following constraints:

4. Graph the constraints given above and indicate the feasible region.

6. Feasible region

7. Andy sells small bags for $3 each and big bags for $5 each. His revenue R (in dollars) is given by R = 3s+5b. Feasible region

8. Andy sells small bags for $3 each and big bags for $5 each. His revenue R (in dollars) is given by R = 3s+5b. (600, 120)

9. (i)Andy should sell 600 small bags and 120 big bags to maximise his revenue. (ii) The maximum revenue is $2400

10. 2004 Question 4 Andy has decided to make up bags containing 200 grams of coloured sweets. The bags contain different mixtures of coloured sweets. Andy has three different colours of sweets in stock. Bags of mixture A contain 100 grams of purple sweets, 50 grams of red sweets and 50 grams of green sweets. Bags of mixture B contain 100 grams of purple sweets and 100 grams of green sweets. Andy has a total of 60 kilograms of purple sweets, 25 kilograms of red sweets and 45 kilograms of green sweets. Andy sells bags of mixture A for $2.45, and bags of mixture B for $1.95.

11. Constraints PurpleRedGreen A100g50 B100g 60kg25kg45kg

12. Feasible region

13. Determine how many bags of each mixture Andy should sell to maximise his revenue, and clearly state his maximum revenue.

14. Objective Function

15. (500, 100)

16. Maximum revenue is $1420 by selling 500 bags of mixture A and 100 bags of mixture B.

17. (b) Andy later decides to sell both mixtures for the same price. Explain fully the effect this decision has on the number of bags of each mixture that he should sell to maximise his revenue.

18. New objective function is

19. The gradient is the same as one of the constraints i.e. This means we get multiple solutions

20. The solutions are whole number values where

21. 2005 Question Three To supplement his income, Marty makes wooden frames for climbing roses, which he then sells to the local garden centres. Marty makes and sells two different-sized frames, a medium one and a large one. Let x represent the number of medium frames Marty makes and sells in a week, and y represent the number of large frames Marty makes and sells in a week. The following inequations represent the weekly constraints in the production of Marty’s frames:

22. Draw these constraints on the axes below, and show the feasible region.

23. Feasible region

24. Marty's weekly profit($P) from the frames is given by the equation Calculate Marty's maximum weekly profit.

25. (56,40)

26. 2005 Question 7 Maria decides she can produce rose frames that are similar to those Marty was making in Question Three, but they will be cheaper. Maria calculates that each medium frame only needs a 6 000 mm length of timber and can be built in four minutes, while a large frame only needs a 12 000 mm length of timber and can be built in six minutes. She has a regular supply of suitable timber, a total length of 900 metres per week. She also has a maximum of eight hours a week to build them. Maria contacts the manager of a local chain of hardware stores. The manager is willing to stock Maria’s frames if she can guarantee at least 20 of each size per week, but will take no more than 60 of each size per week.

27. 2005 Question 7 Maria decides she can produce rose frames that are similar to those Marty was making in Question Three, but they will be cheaper. Maria calculates that each medium frame only needs a 6 000 mm length of timber and can be built in four minutes, while a large frame only needs a 12 000 mm length of timber and can be built in six minutes. She has a regular supply of suitable timber, a total length of 900 metres per week. She also has a maximum of eight hours a week to build them. Maria contacts the manager of a local chain of hardware stores. The manager is willing to stock Maria’s frames if she can guarantee at least 20 of each size per week, but will take no more than 60 of each size per week.

28. Define the variables x = number of medium frames made per week. y = number of large frames made per week.

29. Constraints

30. Feasible region

31. The deal will give Maria a $6 profit on each medium frame produced, and a $10 profit on each large frame.

32. (30, 60)

33. Maximum profit when 30 medium frames are made per week. 60 large frames are made per week For a profit of $780

34. In order to increase her profit, Maria decides she will work more hours each week. All other conditions are unchanged. What is the minimum number of hours she must work to obtain the maximum profit?

35. (60, 45) Move the time constraint as far as possible in the feasible region

36. The maximum profit now is $810

37. 2006 Question 3 Marni makes and sells two types of scented soaps. Her “Vitamin E & Chamomile” soap takes 6 minutes to make, and uses 20 grams of a fat and sodium mix. Her “Aloe & Lanolin” soap takes 5 minutes to make, and uses 30 grams of the fat and sodium mix. Each day Marni has 1500 grams of the fat and sodium mix available, and can work for 330 minutes. She must produce daily at least 15 “Vitamin E & Chamomile” soaps and at least 12 “Aloe & Lanolin” soaps to satisfy existing client orders. However, she has enough customer interest in her products to be able to sell any extra she produces. Let x be the number of “Vitamin E & Chamomile” soaps produced per day and y be the number of “Aloe & Lanolin” soaps produced per day.

38. A linear programming problem for this situation has the following constraints: A: 20x + 30y ≤ 1500 B: 6x + 5y ≤ 330 C: x ≥ 15 D: y ≥ 12

39. Feasible region

40. Marni sells the “Vitamin E & Chamomile” soap for $1.25 each, and the “Aloe & Lanolin” soap for $1.40 each. Her income I ($) is given by the equation I = 1.25x + 1.4y Calculate how many of each soap Marni should make each day in order to maximise her income.

41. (30, 30)

42. Vertices method Identify vertices and value of I (15,12) $35.55 (15,40) $74.75 (30,30) $79.50 (45,12) $73.05 maximum income when 30 of each soap is produced.

43. 2006 Question 6 Marni’s friend Vili has a small business, making sun-shelters and tents for small children. The table below summarises the time it takes to produce each item, and the amount of material it uses.

44. Production time (minutes) Amount of material used (m 2 ) Sun-shelter3 2 Tent405 Let x be the number of sun-shelters produced each week, and y be the number of tents produced each week.

45. Production time (minutes) Amount of material used (m 2 ) Sun-shelter3 2 Tent405 Each week, Vili has available 30 hours to work on making these products, and a total of 190 square metres of material. Currently Vili has a steady order to produce at least 10 sun-shelters and 15 tents each week for a stall at the local market. Any extras he produces can always be sold to a local store.

46. Constraints

47. Objective Function

48. Feasible region

49. (20, 30)

50. Vertex method Vertex P = 8x + 12y (10,15) 260 (10,34) 488 (20,30) 520 (40,15) 500 20 sun shelters and 30 tents for a maximum profit of $520

51. (b) Suppose Vili’s profit for each sun-shelter becomes $9. Explain how this changes the solution obtained in part (a)(ii). Give mathematical reasons to justify your answer.

52. New objective function

53. Gradient is the same as one of the constraints

54. Multiple solutions Use the gradient -3/4 Start at (20, 30) and go up 4 in x and down 3 in y (24, 27), (28, 24), (32, 21), (36, 18), (40, 15) You should identify all solutions whenever possible.