May 2000: Paper 2 #1

The cost of sending documents from Australia to the United States of America using Immediate Courier is recorded in the table below.

Weight

In kilograms

(W )

Cost in

Australian dollars (AUD)

(C )

0.0 < Weight 0.5 20

0.5 < Weight 1.0 30

1.0 < Weight 1.5 40

1.5 < Weight 2.0 50

2.0 < Weight 2.5 56

2.5 < Weight 3.0 62

3.0 < Weight 3.5 68

3.5 < Weight 4.0 74

This means when the weight is between 0.5 and 1 (including 1 but not including 0.5) the cost is 30 AUD.

This means when the weight is between 0 and 0.5 (including 0.5 but not including 0) the cost is 20 AUD.

(a) On graph paper, using a scale of 2 cm to represent 1 kg on the horizontal axis and 2 cm to represent 10 AUD on the vertical axis, draw and label the graph representing this information.

1 2 43

10

80

70

60

50

40

30

20

Doesn't include 0

Exactly 0.5 still cost 20

Exactly 1 still cost 30

Doesn't include 0.5

Continue graphing the function in the same manner, each time "stepping" up when using the next weight interval.

Weight of package (in kg)

Cos

t of

ship

ping

(in

AU

D)

0.0 < Weight 0.5 20

0.5 < Weight 1.0 30

1.0 < Weight 1.5 40

1.5 < Weight 2.0 50

2.0 < Weight 2.5 56

2.5 < Weight 3.0 62

3.0 < Weight 3.5 68

3.5 < Weight 4.0 74

The cost of sending documents from Australia to the United States of America using Speedy Courier is given by the formula

C = 34 + 8W, where C is the cost in Australian dollars,

W is the weight in kilograms.

(b) (i) Copy and complete the table below

W 0.5 1 2 3 4

C 42

To complete the table, plug in the values of W into the equation above.

If W = 0.5, then C = 34 + 8(0.5) = 38

If W = 4, then C = 34 + 8(4) = 66

If W = 2, then C = 34 + 8(2) = 50

If W = 3, then C = 34 + 8(3) = 58

38 50 58 66

(ii) On the same set of axes as in part (a), draw the graph representing Speedy Courier costs.

W C

0.5 38

1 42

2 50

3 58

4 66

1 2 43

10

80

70

60

50

40

30

20

Weight of package (in kg)

Cos

t of

ship

ping

(in

AU

D)

1 2 43

10

80

70

60

50

40

30

20

Weight of package (in kg)

Cos

t of

ship

ping

(in

AU

D)

Thus, a 2 kg package cost 50 AUD to ship using either company.

(i) The weight for which the cost is the same for both companies

(c) Use your graph to find…(Again, this means show the lines on the graph)

(ii) The difference in cost to send a parcel weighing 3.5 kg.

Thus, the difference is

68 – 62 = 6 AUD

68

62

(d) Kim spent 106 AUD to send 2 packages of documents with a combined weight of 5.5 kgs. Given that he used both courier services, calculate the weight of documents he sent through each one.

Since we know there is a limit of 4 kgs for any one package and Kim used both courier services, we can use trial and error to calculate the weight of documents he sent through each one. Let W1 = weight of package sent through Immediate Courier and W2 = weight of package sent through Speedy Courier.

10666404.01.5

11262503.52.0

11458563.02.5

11654622.53.0

11850682.03.5

12046741.54.0

Total Cost

(in AUD)

Cost for W2

(in AUD)

Cost for W1

(in AUD)

W2

(in kgs)

W1

(in kgs)

Thus, Kim sent a 1.5 kg package through Immediate Courier and a 4.0 kg package through Speedy Courier.