Graphics calculator 1. Brand Model 2. Brand Model Attach your SACE registration number label here © SACE Board of South Australia 2020 Essential Mathematics 2020 Question booklet Topic 2: Measurement (Questions 1 to 3) 30 marks Topic 4: Statistics (Questions 4 to 6) 30 marks Topic 5: Investments and loans (Questions 7 to 9) 30 marks • Answer all questions • Write your answers in this question booklet • You may write on pages 8, 15, and 22 if you need more space • Allow approximately 40 minutes for each topic Examination information Materials • Question booklet • SACE registration number label Instructions • Show appropriate working and steps of logic in this question booklet • Use black or blue pen • You may use a sharp dark pencil for diagrams and graphical representations • Approved calculators may be used — complete the box below Total time: 130 minutes Total marks: 90
page 2 of 22 Question 1 (12 marks) The stage for an open-air concert is a trapezoid shape, as shown in the diagram below. The stage needs to be covered with non-slip rubber to ensure the safety of the performers. back front 15.00 m A B C F E D 15.81 m 15.81 m stage (a) (i) Calculate the length AF to two decimal places. (3 marks) The back of the stage (BC ) is 20 metres long. (ii) Using this information and your answer to part (a)(i), show that the front of the stage (AD) is 30 metres long. (1 mark) (iii) Hence, calculate the area of non-slip rubber required to cover the stage. (2 marks) Topic 2: Measurement
page 3 of 22 PLEASE TURN OVER The audience will be seated in front of the stage and the seating area will be surrounded by a fence, as shown in the diagram below. The fence will begin at point A and passes through points W, X, Y, and Z, finishing at point D. 20 m 30 m stage Audience seating area 45 m 60 m 60 m D C Z Y X W A B 17q 125q fence surrounding audience (b) Calculate the length of YZ. (3 marks) (c) (i) Write the lengths of all sides of the fence on the diagram above. (2 marks) (ii) Hence, calculate the total length of the fence surrounding the audience seating area from point A, through points W, X, Y, and Z, to point D. (1 mark) Topic 2: Measurement
page 4 of 22 Question 2 (9 marks) A sports cap is made by sewing eight panels together and fitting a visor. The template for one of these panels is shown below, with measurements taken at regular intervals of 5.9 cm along the length of the panel. visor one panel Source: adapted from © Gomolach | Dreamstime.com 9.6 cm 13.6 cm 11.4 cm 6.7 cm 1.4 cm 5.9 cm (a) Using Simpson’s rule, calculate the approximate area of one panel in square centimetres (cm 2 ). (3 marks) (b) Calculate in square metres (m 2 ) the area of material required to make the eight panels for one cap. (2 marks) Topic 2: Measurement
page 5 of 22 PLEASE TURN OVER (c) (i) Calculate the maximum number of caps (excluding the visors) that can be made from 1m 2 of material. (2 marks) (ii) Discuss one assumption that could affect the reasonableness of your answer to part (c)(i). (2 marks) Topic 2: Measurement
page 6 of 22 Question 3 (9 marks) The diagram below shows a cylindrical rainwater tank installed at a home. The height of the tank is 200 cm and the diameter is 180 cm. 200 cm 180 cm (a) (i) Calculate the radius of the tank in metres (m). (1 mark) (ii) Show that the volume of the tank is approximately 5 m3 . (2 marks) (iii) Convert the volume of the tank to litres (L). (1 mark) Topic 2: Measurement
page 7 of 22 PLEASE TURN OVER The water in the tank is going to be used to flush one toilet. Research shows that, on average, it takes 3.3 L of water to flush the toilet once. (b) (i) If the tank currently contains 4500 L of water, show that there will be 4.17 m3 of water left in the tank after the toilet has been flushed 100 times. (2 marks) (ii) State one assumption made in calculating the volume of the water left in the tank after the toilet has been flushed 100 times. (1 mark) (iii) Calculate the height of the water in metres when the tank contains 4.17 m3 of water. (2 marks) Topic 2: Measurement
page 8 of 22 Topic 2: Measurement You may write on this page if you need more space to finish your answers to any questions in Topic 2. Make sure to label each answer carefully (e.g. 3(a)(i) continued).
page 9 of 22 PLEASE TURN OVER Question 4 (5 marks) A local council conducted a survey of residents in the council area. The survey asked if residents intended to reduce their energy usage by making physical changes to their houses over the next 12 months. Examples of physical changes include the installation of: • solar panels • exterior blinds • double-glazed windows • LED lights. Participation was voluntary and the survey was completed via an online questionnaire or telephone interview. (a) Discuss one error that may occur due to participation in the survey being voluntary. (2 marks) (b) Discuss improvements to the sampling method that may increase the reliability of the results. (2 marks) (c) If the survey included participants living in rental properties, state one reason why their responses may bias the results. (1 mark) Topic 4: Statistics
page 10 of 22 Question 5 (12 marks) An environmental group investigated a possible link between a person’s age and the percentage of rubbish that they recycle. The environmental group surveyed eight people and recorded each person’s age and the percentage of rubbish that they recycled in 1 week. Age (years) Rubbish recycled (%) 31 50 34 40 43 40 47 30 51 20 55 49 63 10 70 5 (a) On the axes below, sketch a scatter plot of the data from the table above. Indicate the scale on each axis. Recycling data Rubbish recycled (%) Age (years) (3 marks) (b) Calculate the coefficient of determination (r 2 ) and state the strength of the relationship between a person’s age and the percentage of rubbish that they recycle. (2 marks) Topic 4: Statistics
page 11 of 22 PLEASE TURN OVER (c) There is one outlier in the data. (i) State the coordinates of the outlier. (1 mark) (ii) The outlier was due to an error. State one possible reason for the error. (1 mark) (iii) Remove the outlier from the data. Using the data with the outlier removed: (1) State the new coefficient of determination (r 2 ) and explain why it may now be reasonable to make predictions from this data. (2 marks) (2) Calculate the equation of the least squares regression line (line of best fit). Fill in the boxes below to complete the equation. y x (percentage of rubbish recycled) (age) (1 mark) (d) Explain why it is unreasonable to extrapolate this data to an 85-year-old person. Use calculations to justify your response. (2 marks) Topic 4: Statistics
page 12 of 22 Question 6 (13 marks) Air quality data are collected and recorded each hour in locations around the world. The air quality data are converted into an air quality index (AQI), which allows the air quality in different locations to be compared. A higher index indicates poorer air quality. Table 1 shows the AQI for Sydney for the first 10 days in December 2018. Table 2 shows the AQI for Sydney for the first 10 days in December 2019. Table 1: AQI for Sydney, December 2018 Table 2: AQI for Sydney, December 2019 AQI December 2018 AQI December 2019 20 110 18 66 46 138 50 153 40 170 40 160 30 165 26 113 34 76 36 57 Source: adapted from ‘Cook and Phillip Sydney East air pollution: real-time Air Quality Index’, viewed 26 May 2020, https://aqicn.org, NSW Office of Environment and Heritage; World Air Quality Index Project (a) Complete Table 3 (correct to one decimal place). Table 3: Statistical measures of air quality in Sydney Statistical measure AQI December 2018 AQI December 2019 mean 34.0 120.8 median range 32.0 standard deviation 42.8 interquartile range (IQR) 84.0 (2 marks) Topic 4: Statistics
page 13 of 22 PLEASE TURN OVER (b) Complete and label the box-and-whisker diagrams below. AQI for Sydney 10 80 90 120 130 140 150 160 170 20 30 40 50 60 70 100 110 AQI December 2019 Q1 = 76 Q3 = 160 AQI December 2018 min. = 18 max. = 50 AQI (4 marks) The AQI is used to advise the population about the health implications of the level of air pollution on any particular day. Table 4 below shows the health advice for a range of AQI readings. (c) (i) On Table 4 below, circle the health advice that would be given to people on the most polluted day recorded in Sydney during the first 10 days of December 2019. (1 mark) Table 4: Air quality and health advice AQI Air pollution level Health advice 0 – 50 Good None 51 – 100 Moderate Active children and adults, and people with respiratory disease, such as asthma, should limit prolonged outdoor exertion. 101 – 150 Unhealthy for sensitive groups Active children and adults, and people with respiratory disease, such as asthma, should limit outdoor exertion. 151 – 200 Unhealthy Active children and adults, and people with respiratory disease, such as asthma, should limit outdoor exertion; everyone else, especially children, should avoid prolonged outdoor exertion. 201 – 300 Very unhealthy Active children and adults, and people with respiratory disease, such as asthma, should avoid all outdoor exertion; everyone else, especially children, should limit outdoor exertion. 300 + Hazardous Everyone should avoid all outdoor exertion. Source: adapted from ‘Cook and Phillip Sydney East air pollution: real-time Air Quality Index’, viewed 26 May 2020, https://aqicn.org, NSW Office of Environment and Heritage; World Air Quality Index Project Topic 4: Statistics
page 14 of 22 (ii) (1) Tick the appropriate box. Air quality in Sydney was most consistent in: December 2018 December 2019 (1 mark) (2) State one appropriate statistical value that supports your response to part (c)(ii)(1). (1 mark) A newspaper claimed that ‘the air quality in Sydney is getting worse every year’. (d) (i) Using statistical values, state whether or not this claim is supported by the data collected. (2 marks) (ii) Based on the data collected, discuss the reasonableness of this claim. (2 marks) Topic 4: Statistics
page 15 of 22 PLEASE TURN OVER Topic 4: Statistics You may write on this page if you need more space to finish your answers to any questions in Topic 4. Make sure to label each answer carefully (e.g. 6(d)(i) continued). Topic 5: Investments and loans begins on page 16.
page 16 of 22 Question 7 (9 marks) Arjun has found his dream home and needs a bank loan of $350 000. He will repay the loan over 20 years at an interest rate of 4.2% per annum, compounded monthly. (a) Show that Arjun’s minimum monthly repayments are approximately $2160. (2 marks) (b) Calculate the interest paid on the loan over 20 years. (2 marks) Arjun can only afford repayments of $1890 per month. (c) Calculate how long (in years) it will take Arjun to repay the loan with repayments of $1890 per month. Assume that the other loan conditions remain the same. (3 marks) Topic 5: Investments and loans
page 17 of 22 PLEASE TURN OVER (d) Other than increasing the repayment, state two ways in which Arjun could pay off the loan sooner. (2 marks) Topic 5: Investments and loans
page 18 of 22 Question 8 (9 marks) Mylor Bank offers term-deposit investments to its customers, where the full interest earned by the investment is paid at the end of the term. The table below shows the interest rates for all the term-deposit investments that the bank offers. Term More than $10 000, up to $20 000 More than $20 000, up to $50 000 More than $50 000, up to $100 000 More than $100 000 Up to 1 month 5.00% 5.05% 5.20% 5.25% More than 1 month, up to 3 months 5.25% 5.25% 5.50% 5.50% More than 3 months, up to 6 months 5.25% 5.30% 5.50% 5.60% More than 6 months, up to 12 months 5.30% 5.60% 5.95% 5.95% More than 12 months, up to 2 years 5.30% 5.60% 5.95% 6.00% (a) State why an amount of $6500 cannot be invested in a term deposit at Mylor Bank. (1 mark) (b) Dillon has $52 000 to invest for a 6-month term. Calculate the interest that Dillon would earn if he invested this money in a Mylor Bank term deposit. (2 marks) Dillon thinks that he may want to access his money before 6 months has elapsed. He decides to invest the $52 000 in a Mylor Bank term deposit initially for a 3-month term. (c) (i) Show that the interest earned on Dillon’s investment for the 3-month term is $715. (1 mark) (ii) Calculate the total amount that Dillon will have after investing his money for 3 months. (1 mark) Topic 5: Investments and loans
page 19 of 22 PLEASE TURN OVER Dillon decides to reinvest the total amount, calculated in part (c)(ii), for a further 3-month term. (iii) Calculate the final amount that Dillon will have after the second 3-month term. (2 marks) (d) Explain why Dillon earns more interest when investing the money for two 3-month terms rather than one 6-month term, even though these terms have the same interest rate. (2 marks) Topic 5: Investments and loans
page 20 of 22 Topic 5: Investments and loans Question 9 (12 marks) Annie started working for her employer at 18 years of age. Her employer contributes $9500 into Annie’s superannuation account each year. Annie’s fund earns 6.55% per annum, compounded quarterly. (a) (i) Calculate how much the employer pays into Annie’s superannuation account each quarter. (1 mark) (ii) Show that Annie’s superannuation account at age 30 years has a balance of approximately $171 200, assuming that quarterly contributions were made. (2 marks) Annie took 2 years unpaid leave from her workplace when she turned 30 years of age. During this time, her superannuation account continued to earn 6.55% per annum, compounded quarterly; however, she made no contributions to the fund. (b) Show that Annie’s superannuation balance at the end of the 2 years unpaid leave was approximately $195 000. (2 marks)
page 21 of 22 PLEASE TURN OVER Topic 5: Investments and loans Barrie ― Annie’s twin brother ― started working when he turned 25 years of age. He joined the same superannuation fund, earning 6.55% per annum, compounded quarterly. (c) Calculate the quarterly contributions that would allow Barrie to reach the same superannuation balance as Annie when they are both 32 years of age. (2 marks) Barrie will retire when he has $1 000 000 in his superannuation account. (d) Calculate at what age Barrie can retire with $1 000 000 in his superannuation account, if his quarterly contribution from age 25 years is $2375. (3 marks) (e) Explain one factor, other than unpaid leave, that may delay Barrie accumulating $1 000 000 in his superannuation account. (2 marks)
page 22 of 22 — end of booklet You may write on this page if you need more space to finish your answers to any questions in Topic 5. Make sure to label each answer carefully (e.g. 9(a)(i) continued). Topic 5: Investments and loans