Year 12
Mathematics Extension 2
HSC Assessment Task 2, Part A - Assignment
Wednesday 26th February 2020
Deadline: 9:10 am, Wednesday 4th March 2020 (period 2)
General Instructions



Responses should be handwritten using
black pen
Graphs and spreadsheets may be printed
if necessary. Spreadsheets should be
annotated with the formulae used in cells
Show relevant mathematical reasoning
and/or calculations

Write your examination number on each
page handed in

Pages should be numbered and stapled,
with clear headings indicating the
question
Total marks - 17
Question 1
Financial Mathematics (5 marks)
Pages 2-4
Question 2
Volumes of Revolution and rates (12 marks)
Pages 5-7
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Question One:
Throughout this question you should use spreadsheets to help you calculate the relevant quantities, as well as
data given in Figures 1-2 (electronic versions of these can be found on Lampada). Each answer should be
accompanied with two extracts of an annotated spreadsheet, one showing the numbers and the other in ‘Show
Formulae’ view. It should communicate clearly to the examiner what you have done.
a)
r
c
c
A person placed $10 000 at the start of February 2001 into a bank account that has a variable interest
rate that compounds biannually (i.e. every 6 months). The interest rate is given by the average of the
cash rate set by the Reserve Bank of Australia over those 6 months, and the per annum cash rate values
can be found in Figure 1. Assume that the rate set in December applies through January too.
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If they didn’t deposit nor withdraw any money, show that the first 6 month period in which there would
b be at least $20 000 in the bank account begins in February 2018.
b) Each year, inflation reduces the spending power of a given amount of money, as items cost more to buy.
U Using the data for historical inflation rates given in Figure 2, work out what amount of money at the start
o of 2001 would have been worth $20 000 at the end of 2017 through inflation.
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c) Taking inflation into account, as well as the interest earned on the account, does the bank account in
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h part a) have more or less spending power near the start of 2001 or at the end of 2017? Explain your answer.
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Figure 1. Historical data on the cash rate provided by the RBA, which can be accessed through the link given in
the top left cell. Note that the rates are given every month except for January, in which you can assume the rate
stays the same as in the month prior to it. Use this data for 1a) and c).
Note that this data is in the Excel file ‘2020 12ME2-1 HSC Assessment Task 2 Part A’ under the tab ‘Figure 1’
on Lampada, and you should use the spreadsheet to carry out any calculations.
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Figure 2. Historical data on inflation rates. Use this data for 1b) and c).
Note that this data is in the Excel file ‘2020 12ME2-1 HSC Assessment Task 2 Part A’ under the tab ‘Figure 2’
on Lampada, and you should use the spreadsheet to carry out any calculations.
End of Question One
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Question Two:
For parts a), b) and c)(i) of this question normal handwritten responses should be submitted. In part c) (ii) and
(iii) you should submit print outs of the relevant graphs, showing any formulae clearly.
a) Water is poured into a cylinder with height H cm and radius R cm, at a constant flow rate of
dV
 Q cm3s-1. The water has height h cm above the bottom of the cylinder at time t seconds, and the
f
dt
c cylinder is initially empty. Find the rate of change of the height of the water in terms of Q and R.
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b) A glass with a parabolic vertical cross section and circular horizontal cross section, as in Figure 3 below,
s is being filled with water. The height of the liquid from the bottom of the glass after pouring water into
I it from empty after t seconds is given by h cm, the radius at the top of the glass is R cm, and the maximum
h height of the water before the glass overflows is given by H cm. The volume flow rate is constant and is
g given by Q cm3s-1.
R
r
r
H
h
Figure 3. Top left is a 3 dimensional view of the glass. Only the top part that holds liquid is relevant
here. Middle is the vertical cross section, which is a parabola. The height h is defined upwards from
the centre of the bottom of the glass. Top right shows a horizontal cross section which is a circle.
(i)
By using volumes of revolution and choosing the origin at the centre and bottom of the glass
with the side of the glass being given by equation y  k x 2 , find an expression for the exact
volume V cm3 of water in the glass at a given moment in terms of h and k.
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(ii)
Find the exact rate of change of the height of the water in the glass as a function of time,
flow rate Q and k.
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(iii)
If the glass has total height H = 8, largest radius R = 6 cm, and Q = 10 cm3s-1, what is the rate
of change of the height of the water at t = 3 seconds? Give your answer to one decimal place.
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c) Geogebra can be used to fit functions such as polynomials to a set of points. For example, using the FitPoly
c command:
FitPoly[{List of points separated by commas}, degree of polynomial]
a
s
F
b
p
h
p
To test this, create points with coordinates A=(0,0), B=(1,2), C=(2,4), D=(3,3), E=(5,4), F=(7,8) and create
an equation FitPoly[{A,B,C,D,E,F}, 2]. This fits a polynomial with degree 2 to the points. As you should
see, it is a model that doesn’t fit the points very well. Modify the equation so that it now says
FitPoly[{A,B,C,D,E,F}, 3]. This fits a polynomial with degree 3. Similarly by changing the last number
before the brackets to 4 or 5 it fits polynomials with higher degrees. You may wish to consider why the
polynomial of degree 5 goes through every point exactly but that the program cannot plot polynomials of
higher degree than this for this number of points. Geogebra can be used to fit other functions than
polynomials to points also. This link gives more detail on this:
https://wiki.geogebra.org/en/Fit_Command
Additionally, Geogebra can be used to find (and visualise) volumes of revolution. You may use this link to
d do this:
https://www.geogebra.org/m/mzWq2Cet
This allows us to approximate volumes of glasses by fitting functions that closely match the dimensions of
t the glass and then calculating the volume of revolution.
(i)
Given in figure 4 overleaf is a diagram of a glass overlaid on a grid, with a scale. Assume that
its horizontal cross section is circular throughout. Explain why if you take the origin at the
centre of the bottom of the glass (indicated by an X), you would have to turn it on its side in
order to fit a function to the line tracing the side of the glass.
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(ii)
Plot points corresponding to its side on geogebra and fit an appropriate function to represent
it. Paste a picture of your function, showing the formula and coordinates of the plotted points.
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(iii)
Use the link given above on finding volumes of revolution with your function found in part (ii)
to find the approximate capacity of the glass, and give your answer to the nearest 10 cm3. You
should paste a picture of the volume of revolution used.
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Figure 4.
End of Assignment
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