MUF0141 Linear Programming Exam Questions PDF

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MUF0141 Fundamental Mathematics Unit 1

Linear Programming Exam Questions Question 1 A calculator manufacturer decides to stop production of two models of calculator, X and Y, when existing parts have been used. Stock of screws, clips and batteries are low, and the requirements for each calculator produced are as follows.

a.

Write down the constraints on x and y

b.

Indicate on the grid the region for which x and y satisfy the constraints.

c.

The profit for calculator X and Y are $12 and $10 respectively. Find the number of each calculator to be sold to maximise the profit.

d.

Extra batteries are readily available. How many extra batteries should the manufacturer buy so that all the screws and clips can be used? How many X and Y calculators can then be made?

MUF0141 Fundamental Mathematics Unit 1 Question 2

A manufacturer makes training shells for both rowing and sculling. The manufacturer wants to determine the optional number of each type of shell he should make in order to make the maximum profit. The manufacturer has identified the following constraints on his production:

• He cannot make more than a total of 40 shells of either type each month • Due to limits on materials he cannot make more than 20 rowing shells each month • Oars are supplied to the manufacturer and he cannot obtain more than 100 oars from his supplier each month. Each rowing shell requres 4 oars and each sculling shell requires 2 oars.

The number of rowing shells are x and the number of sculling shells are y. The inequations for each of the constraints listed above are:

a.

Show each of the regions listed above on a grid clearly indicating the equations of each line that is drawn, the feasible region and the corner points associated with the feasible region.

b.

The manufacturer makes a $600 profit on a rowing shell and $500 profit on a sculling shell. Write down the objective function representing this profit.

c.

Use the corner points of the feasible region to calculate how many of each shell should be made in order to maximise the profit.

d.

What is the maximum profit that this manufacturer can make each month?

e.

If the profit on a rowing shell decreases to $300, what is the new objective function?

f.

What is the maximum profit that the manufacturer can now make and how many of each shell should he make to maximise his profit?

MUF0141 Fundamental Mathematics Unit 1 Question 3 The hiring company has a fleet of small and large trucks which are used to deliver equipment and furniture. Each small truck requires two people on board and each large truck requires three people. On a particular day, there are only 10 people who can go out on the trucks. Also on a particular day, there are only 4 small trucks and 3 large trucks that are available for deliveries. Let x be the number of small trucks out delivering in a day. Let y be the number of large trucks out delivering in a day. The constraints on the number of trucks out delivering are given by:

a.

Show the feasible region on your graph and find how many points on the graph satisfy the constraints of

small and large trucks that could be out delivering on this particular day.

For each small truck that is out delivering, the company can make a profit of $3 400 a

day.

For each large truck the profit can be $5 200. b.

Find the maximum profit that can be made on this particular day. A further limitation on the use of the trucks occurs because of the fuel shortage. On average, a small truck uses 125 litres of fuel per day and a large truck uses 250 litres. The company has

c.

access to 500 litres of fuel a day.

How many small and large trucks should be out delivering in a day to maximise profit during this fuel shortage?

MUF0141 Fundamental Mathematics Unit 1 Question 4

(9 marks)

The unshaded region shown in the graph below is the feasible region for a linear programming problem.

There are four constraints acting on the variables x and y. One of the constraints is shown in the graph above. a.

State the other three constraints.

(3 marks)

MUF0141 Fundamental Mathematics Unit 1 b.

(i)

On the grid below, graph the solution to the system of linear inequations given by:

x + 2y  6 y > 2x - 4 x > -1 Clearly label axis intercepts. Working space is provided below to find axis intercepts. You do not need to find the corner points.

(5 marks)

(ii)

Shade in the feasible region corresponding to this set of inequations.

(1 mark)

MUF0141 Fundamental Mathematics Unit 1 Question 5

(18 marks)

Ballarat Couriers (BC) delivers small (under 10kg) and large (over 10kg) parcels. There are staff and transport constraints on the number of parcels BC can deliver in a day. Constraint 1:

5 x  8 y 1600

(due to transport limitations).

Constraint 2:

x  y 260

(due to staff limitations).

x:

the number of small parcels delivered in one day.

y:

the number of large parcels delivered in one day.

Furthermore, in a day BC, must deliver at least 50 large parcels and between 20 and 80 (inclusively) small parcels. These constraints (3, 4 and 5) are shown on the graph below.

a.

State the constraints 3, 4 and 5 that are shown in the graph below.

(3 marks)

MUF0141 Fundamental Mathematics Unit 1 b.

The graph from the previous page is repeated below. On this graph, sketch constraints 1 and 2. Label each constraint.

c.

(4 marks)

(i)

On the graph above, shade the feasible region.

(1 mark)

(ii)

Find each of the corner points of the feasible region and label them on the graph

above. You do not need to show working.

(4 marks)

Ballarat Couriers (BC) makes a profit of $3.50 for delivering a small parcel and $5 for delivering a large parcel. d.

(i)

Write an equation for the profit P ($) that BC makes for delivering small and large parcels.

(ii)

Calculate the maximum profit that BC can make in one day for delivering small and large parcels. You must show your working.

(iii)

(2 marks)

(3 marks)

How many small parcels must be delivered in one day to achieve this maximum profit?

(1 mark)

MUF0141 Fundamental Mathematics Unit 1 Solutions Question 1 a.

b.

c.

Profit = 12x +10y From the graph vertices are: (0,80) Profit $1800 (90,35) Profit $1430 (110,15) Profit $1470 (120,0) Profit $1440 To maximise the profit, the company needs to produce 110 units of calculator X and 15 units of calculator Y.

MUF0141 Fundamental Mathematics Unit 1 d.

This changes the feasibility region of the graph (below). The company must buy 5 additional batteries to use all clips and screws. 100 units of calculator X and 30 units of calculator Y can then be made.

MUF0141 Fundamental Mathematics Unit 1 Question 2 a.

Graph

b.

P = 600× x + 500× y

c.

(0, 0) Profit $0 (0, 40) Profit $2000 (10, 30) Profit $21000 (20, 10) Profit $17000 (20, 0) Profit $12000 Maximum

d.

Maximum profit is $21 000

e.

P = 300x + 500y

f.

(0, 40) Profit $20000 (10, 30) Profit $18000 (20, 10) Profit $11000 Maximum profit $20 000 if making 40 sculling shells only.

Question 3 a.

Graph

b.

P = 3400x + 5200 y (3, 1)Profit $15400 (2, 2) Profit $17200 ⇒ Max Profit (0, 3) Profit $15600 (4, 0)

c.

125x + 250 y ≤ 500 P = 3400x + 5200 y (1, 1) Profit $8600 (0, 2) Profit $10400 (4, 0) Profit $13600 ⇒ 4 small trucks and 0 large trucks for max profit.

MUF0141 Fundamental Mathematics Unit 1 Question 4 a.

x 0 y 0 Accept x > 0; y > 0.

y -3x+15

b.

MUF0141 Fundamental Mathematics Unit 1 Question 5 a.

Constraint 3: x  20 Constraint 4: x 80 Constraint 5: y 50

b.

c.

corner points are: (20, 50) (80, 50) (80, 150) (20, 187.5)

MUF0141 Fundamental Mathematics Unit 1 d. (i)

P=3.5x+5y

d. (ii)

d. (iii)

The maximum profit made in one day = $1030...