VCE General Maths Data Analysis 2019 Mini Test 3

Question 1 (3 marks) [2019 VCE Further Mathematics Exam 2 Q6]

The total rainfall, in millimetres, for each of the four seasons in 2015 and 2016 is shown in Table 5 below.

Table 5

Year	Total rainfall (mm)
	Summer	Autumn	Winter	Spring
2015	142	156	222	120
2016	135	153	216	96

a. The seasonal index for winter is shown in Table 6 below.
Use the values in Table 5 to find the seasonal indices for summer, autumn and spring.
Write your answers in Table 6, rounded to two decimal places. 2 marks

Table 6

	Summer	Autumn	Winter	Spring
Seasonal index			1.41

b. The total rainfall for each of the four seasons in 2017 is shown in Table 7 below.

Table 7

Year	Total rainfall (mm)
	Summer	Autumn	Winter	Spring
2017	141	156	262	120

Use the appropriate seasonal index from Table 6 to deseasonalise the total rainfall for winter in 2017.
Round your answer to the nearest whole number. 1 mark

Recursion and financial modelling

Question 2 (4 marks) [2019 VCE Further Mathematics Exam 2 Q7]

Phil is a builder who has purchased a large set of tools.
The value of Phil’s tools is depreciated using the reducing balance method.
The value of the tools, in dollars, after \(n\) years, \(V_n\), can be modelled by the recurrence relation shown below.

\(V_0 = 60\,000, \quad V_{n+1} = 0.9 V_n\)

a. Use recursion to show that the value of the tools after two years, \(V_2\), is $48 600. 1 mark

b. What is the annual percentage rate of depreciation used by Phil? 1 mark

c. Phil plans to replace these tools when their value first falls below $20 000.
After how many years will Phil replace these tools? 1 mark

d. Phil has another option for depreciation. He depreciates the value of the tools by a flat rate of 8% of the purchase price per annum.
Let \(V_n\) be the value of the tools after \(n\) years, in dollars.
Write down a recurrence relation, in terms of \(V_0\), \(V_{n+1}\) and \(V_n\), that could be used to model the value of the tools using this flat rate depreciation. 1 mark