## Linear Programming Exercise

1. Draw a graph of the follow sets of inequalities, shading OUT the required regions:

a. 3 ≤ x < 5
1 < y ≤ 4

b. x + y ≤ 8
y < 2x + 7
y > -3

c. 3y + 5x ≤15
x + 2y < 4
x ≥ 0
y ≥ 0

2. Louise makes and sells two types of herbal hand-cream: rosemary and lavender. Each jar of rosemary cream takes 4 minutes to make and requires 60 grams of lanolin. Each jar of lavender cream takes 12 minutes to make and requires 100 grams of lanolin. Each day Louise can spend at most 180 minutes making hand-cream and can obtain up to 1800 grams of lanolin. Her friend Camille does the packing and can pack at most 25 jars a day.

a. Let x represent the number of jars of rosemary cream and y represent the number of jars of lavender cream. The boundary lines of the main constraints on x and y have been drawn on the graph below. On the copy of the graph below, add the labels "time", "lanolin" and "packing" to the appropriate constraint lines, and clearly indicate the feasible region. b. The rosemary cream sells at a profit of \$4 a jar and the lavender cream sells at profit of \$5 a jar. Find the solution to the linear programming problem that maximize the total profit.

c. Suppose that Louise can make only whole numbers of jars each day. Given that you need to consider only the four integer points neighbouring the solution found in part b, find how many jars of each cream should be made to maximise the total profit.

3. The following graph shows three straight lines: a. Give the four inequalities that describe the shaded region.

b. List the coordinates of the four vertices A, B, C and D, of the shaded region.

c. Find the maximum value of 5x + 4y on the shaded region. Show your working.

4. Kathy has a cheque account with her bank and is concerned about her bank fees. Her bank has one fee for paper transactions (cheques and over-the-counter withdrawls and deposits) and a different fee for electronic transactions (telephone , EFTPOS and ATM transactions).

Let x represent the number of paper transactions Kathy makes in a month and y represent the number of electronic transactions she makes in a month.

Kathy looks at her bank statements for the previous year and identifies four constraints on the number of transactions. The table shows each constraint in words and as an inequation. The table is not complete.

 Constraints Number In words Inequation 1 There are at least 4 paper transactions in a month. x ≥ 4 2 Two times the number of paper transactions plus the number of electronic transactions is at least 18. 2x + y ≥ 18 3 y ≥ x 4 There are, at the most, 12 electronic transactions per month.

a.       (i) Express constraint 3 in words.

(ii) Write an inequation for constraint 4.

b. The boundary lines for constraint 1, constraint 2 and constraint 3 have been drawn on the graph.

(i) Draw the boundary line for constraint 4.

(ii) Clearly identify the feasible region which satisfies the system of inequations representing these four constraints. c. Each paper transaction costs 40 cents and each electronic transaction costs 25 cents. Write an expression for the total monthly bank fee in cents for Kathy's transactions.

d. Find the number of paper transactions and the number of electronic transactions so that Kathy's total bank fee for transactions is a minimum. Show your working.