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Mathematical Methods (CAS) Unit 3 SAC: Application Task Name: _______________________________

Teacher: __________________________________

This component of the Application Task is completed over three periods under SAC conditions. It is submitted to your teacher at the end of each class, to be returned at the start of the next. You may use your textbook, summary book or class notes in completing this task. _______________________________________________________________________________________________ If you’ve ever been on a roller-coaster, you’ll know the thrills (or terror!) of being hauled high into the air in a tiny cart on wheels, only to be let go and hurled towards the ground again. Modern roller-coasters go beyond simple ups and downs, turning upside down, running through spirals and other terrifying manoeuvres. The track of a roller-coaster is an ideal subject for mathematical modelling. Some features, like a cork-screw or a loop, are complex and beyond the scope of Mathematical Methods Unit 3, but we can use our knowledge of calculus and functions to model a simple roller-coaster track and that’s exactly what this SAC is all about. Below is a plan for a simple roller-coaster. This roller-coaster is made up of three stages and we’ll do some analysis on each of these stages separately. y

The track is drawn on a set of Cartesian axes. The horizontal axis represents distance from the starting point while the vertical axis represents the height above the ground. We’ll assume that the ground is flat and that the horizontal axis is at ground level. x

Our aim in this SAC Application Task is to find a mathematical model for the entire roller-coaster track.

Instructions: Answer all questions in the spaces provided. In all questions where a numerical answer is required, an exact value must be given unless otherwise specified. In questions where more than one mark is available, full marks may not be awarded if appropriate working is not shown.

Total Marks = 49

1

Part A Stage One of the rollercoaster has three sections to it as shown in the diagram below. y

D (50,30)

30

Section 3 20

C (40,15)

10

A (0,2) 10

Section 2

B (30,2)

Section 1 20

30

40

50

x

-10 -20

1. Calculate the average rate of change between points A and C ______________________________________________________________________________________ _____________________________________________________________________________________ 1 mark 2. Section 1 of the roller-coaster track joins point A and B and is a simple flat piece of track that is 30m long. This section of the graph sits 2m above the ground. Call this section of the graph f1(x). Write down the equation for f1(x). __________________________________________________________________________________ 1 mark 3. Section 2 joins points B and C and is modelled with a quadratic function. Call this section of the graph f2(x). a. Explain why the stationary point must be at point B if section 2 joins smoothly with section 1. ________________________________________________________________________________________ ________________________________________________________________________________________ __________________________________________________________________________________1 mark b. Use the general formula for a quadratic in turning point form, f2(x) = a (x - b)2 + c, to write down the value of b and c. ________________________________________________________________________________________ _________________________________________________________________________________ 2 marks c. Section 2 ends at point C. Determine the value of a and write the equation for f2(x). ________________________________________________________________________________________ ________________________________________________________________________________________ _________________________________________________________________________________ 2 marks

2

Section 3 joins points C and D and is also modelled with a quadratic function with zero gradient at D. Call this section of the graph f3(x). 4a. What are the coordinates of the stationary point of f3(x)? __________________________________________________________________________________ 1 mark b. Determine the equation for f3(x). ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ _________________________________________________________________________________ 2 marks c. Verify that section 2 and section 3 do not join smoothly at point C. ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ _________________________________________________________________________________ 3 marks d. Section 3 could be modelled by a function of the form f (x )  a log e (x  h )  k passing through points C and D but without a gradient of zero at point D. Again sections 2 and 3 do not join smoothly. Assume that f(x) is the image after loge(x) has been mapped from its start point (1,0) to point C. Determine the values of a, h and k ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ _____________________________________________________________________________

4 marks

e. The inverse of a logarithm function could be used to model section 2.

x 2

Determine the inverse function of f (x )  log e ( ) 5 ________________________________________________________________________________________ ________________________________________________________________________________________ _________________________________________________________________________________ 2 marks 3

The engineers decide to use a cubic function to model Section 2 and Section 3 as one section. This section must pass through points B, C and D and must join smoothly at point B. 5. Call this combined section of the graph f23(x). Use the general cubic rule f23(x) = ax3 + bx2 + cx + d to model this section of the graph. a. This section of the track passes through the points B, C and D. Write down three equations for this information. ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ _________________________________________________________________________________ 3 marks b. The gradient of this section of the track must be zero at the point B. Write down an equation for this information. ________________________________________________________________________________________ ________________________________________________________________________________________ _________________________________________________________________________________ 1 mark

c. Hence or otherwise, write down the equation for f23(x) using exact coefficients. ________________________________________________________________________________________ _________________________________________________________________________________ 1 mark

d. Verify that this rule is not a suitable model for this section of the track because it does not have zero gradient at the point D. ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ _________________________________________________________________________________ 2 marks

4

6. Determine a cubic model for the section of the roller-coaster track that joins point B and D with zero gradient at each end. This model does not pass through point C. ___________________________________________________________________________________________ ___________________________________________________________________________________________ ___________________________________________________________________________________________ ___________________________________________________________________________________________ ___________________________________________________________________________________________ ___________________________________________________________________________________________ ___________________________________________________________________________________________ ____________________________________________________________________________________ 3 marks

5

Name:_________________________________

Teacher:___________________

Part B Stage Two of the plan consists of one section of track as shown in the diagram below. y 40 D (50,30)

30 20

E (70,k)

10

F (80,0) 10

20

30

40

50

60

70

80

90

x

-10 -20

This section of the track is modelled with another cubic function. It passes through the points D, F and another point E with coordinates (70,k). This section of the graph must join Stage One smoothly at the point D. 1. Call this section of the graph f4(x). a. Use the general cubic equation y = ax3 + bx2 + cx + d and the points D, E and F to write down four (4) equations that can be used to determine the rule for f4(x). One of these equations will be in terms of k. ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ _________________________________________________________________________________ 2 marks b. Hence or otherwise, write down the equation of the function f4(x) in terms of k. ________________________________________________________________________________________ ________________________________________________________________________________________ _________________________________________________________________________________ 2 marks 2. Using the equation of function f4(x) determined above: a. Write down an equation for f4'(x), the gradient of this section of the graph at any point, in terms of k. ________________________________________________________________________________________ __________________________________________________________________________________ 1 mark b. If k = 1.4, the gradient of the curve at 60m along the track is -1.8 to one decimal place. Use this information to confirm your answer to f4'(x) above is correct. ________________________________________________________________________________________ ________________________________________________________________________________________ _________________________________________________________________________________ 2 marks 6

3. Stage Two of the track ends at a point G. 110 a. Write down the equation of the function f4(x) if k = 9 . ________________________________________________________________________________________ __________________________________________________________________________________2 marks b. The gradient of the track at point G is again zero. Show that x = 90 at point G for f4(x) above. ________________________________________________________________________________________ __________________________________________________________________________________2 marks c. State the coordinates of point G. __________________________________________________________________________________1 mark

Part C Stage Three of the track was planned to start at point G and have zero gradient at this point. The track rises dramatically into the air from this point and terminates at a height of 50m at a point that is less than110m from the starting point of the track. The roller coaster will eventually come to a momentary stop at this point before reversing its motion back to the starting point. The diagram below shows Stage Three of the track.

1. It is decided to use a hyperbolic function to model the last section of the track. a. The engineers realise that they must extend Stage Two of the track, f4(x), through the point G if they are to use a hyperbolic model for Stage Three. Explain why this is so. ________________________________________________________________________________________ __________________________________________________________________________________ 1 mark

7

110

b. Stage Two of the track, f4(x) when k = 9 , is extended through Point G to a point where the gradient is 0.5 to create Stage Three. Stage Three must end at a point that is less than 110m horizontally from the starting point. Stage Three of the track must join Stage Two smoothly. The model for Stage Three will have the rule

f5(x) =

A +c x – 110

Determine the equation and domain for f5(x), Stage Three of the track, correct to 3 decimal places. ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ ________________________________________________________________________________________ _________________________________________________________________________________ 7 marks

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