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UNIT 4 Mathematical Methods Modelling / Problem solving task (SAC 2)

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UNIT 4 Mathematical Methods, 2019 Modelling / Problem solving task (SAC 2) Question 1: (35 marks)- DESIGNING & CONSTRUCTING THE SWIMMING POOL Highvale Secondary College is proposing to build an aquatic centre with a large swimming pool. In order to build the pool, the builders must excavate (dig into the ground) to remove the soil. For this entire SAC, lets assume that the ground is level, and can be shown by the positive x-axis. The architect in charge of this project has proposed two possible excavation models for the pool. These are shown below:   3 6 2 : 0,50 where ( ) 2 6250 125 x x p R p x → = - - 3 2 :[0,50] where ( ) 2 25 x j R j x → = - - a. Analyse both functions to determine the following: (write solutions in the table below) i. ii. iii. iv. | function minimum function maximum function range point(s) of intersection (approximate to 1 decimal place) | 1x4 marks= 4 marks function minimum function maximum function range point(s) of intersectionb. Sketch the derivative graphs p x j x '( ) and '( ) for both the functions in the given graphing grid and label all key points ( two decimal places, where approximation is required) 3 marks UNIT 4 Mathematical Methods, 2019 Modelling / Problem solving task (SAC 2) Page 2 c. Using Calculus obtain the derivatives p x '( ) and j x '( ) for the functions p x j x ( ) and ( ) 2x2 marks= 4marks d. What information do the two intersection points of the functions p x '( ) and j x '( ) tell us? 2 marks e. i. Using calculus find the area as an exact value between the function p x x ( ) and the -axis 2 marks ii. Using calculus find the area as an exact value between the function j x x ( ) and the -axis 2 marks UNIT 4 Mathematical Methods, 2019 Modelling / Problem solving task (SAC 2) Page 3 iii. What do you conclude from your answers to parts i. and ii. above? 1 mark f. i. Find the average value for the function p(x). ii. What information does this average value convey? | 1 mark 1 mark i. ii.g. i. By analyzing all the answers from parts a-e above, which of the two models will you recommend for excavation? Justify you answer. 2 marks UNIT 4 Mathematical Methods, 2019 Modelling / Problem solving task (SAC 2) Page 4 ii. Using Calculus obtain the area bound between the functions p(x) and j(x) given that the approximate x-coordinate expressed as a fraction for the point of intersection of the two functions is . (Answer correct to three decimal places ) | 2 marks19.1 or 191 10 x  (After stating the actual Integral / rule with correct limits, you may use a calculator to approximate values while performing the area calculations) iii. What conclusion can be drawn about the two individual sectional areas created by the point of intersection of the functions p(x) and j(x), Use CAS to answer this question? 2 marks h. Given that the cost of excavation is $ 80 per cubic meter ( m3 ). The width of the pool, which has to be fitted in the excavated hole, is 25 m. Determine the total cost for the excavation using the function you chose as your preferred mode

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