NCEA Questions

Question 3 2004Andy’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.

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

Graph the constraints given above and indicate the feasible region.

Feasible region

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

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)

(i) Andy should sell 600 small bags and 120 big bags to maximise his revenue.

(ii) The maximum revenue is $2400

2004 Question 4Andy 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.

ConstraintsPurple Red Green

A 100g 50 50B 100g 100g

60kg 25kg 45kg

Feasible region

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

Objective Function

(500, 100)

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

(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 shouldsell to maximise his revenue.

New objective function is

The gradient is the same as one of the constraints i.e.

This means we get multiple solutions

The solutions are whole number values where

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:

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

Feasible region

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

Calculate Marty's maximum weekly profit.

(56,40)

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.

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.

Define the variables

x = number of medium frames made per week.y = number of large frames made per week.

Constraints

Feasible region

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

(30, 60)

Maximum profit when30 medium frames are made per week.60 large frames are made per week For a profit of $780

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?

(60, 45)

Move the time constraint as far as possible in the feasible region

The maximum profit now is $810

2006 Question 3Marni 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.

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

A: 20x + 30y ≤ 1500B: 6x + 5y ≤ 330 C: x ≥ 15D: y ≥ 12

Feasible region

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.

(30, 30)

Vertices method

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

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.

 Production time (minutes)

Amount of material used

(m2)Sun-shelter 3  2Tent 40 5

Let x be the number of sun-shelters produced each week, and y be the number of tents produced each week.

 Production time (minutes)

Amount of material used

(m2)Sun-shelter 3  2Tent 40 5

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.

Constraints

Objective Function

Feasible region

(20, 30)

Vertex method

Vertex P = 8x + 12y (10,15) 260(10,34) 488(20,30) 520(40,15) 50020 sun shelters and 30 tents for a maximum profit of $520

(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.

New objective function

Gradient is the same as one of the constraints

Multiple solutions

Use the gradient -3/4Start 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.