Question

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3 Rally-car prototyping [5 marks] You have just started a new role as an automotive engineer, your job is to design rally cars and analyse their performance. You have information about the time taken for a rally car prototype to complete three different courses, each with three types of terrain; sealed road, gravel road and snow/ice road. You also know the total distance travelled by the car during the tests. From this information you need to determine the average speed of the prototype on each type of terrain. The information available to you is summarised in the following table: Course Distance, sealed Distance, gravel [kmDistance, snow ice 90 80 110 200 180 190 130 160 145 Total time 101/20 (5.05) 108/'20 (s 5.4) 107/20 (5.35) (a) Using this information, construct a system of three linear equations with three unknowns, r, y and z, and describe what the quantities r, y and z, related to the unknown average speeds, represent [1 mark] Hint: you will need to appropriately rearrange the relationship average speed distance / time. (b) By forming an appropriate matrix system and applying Gaussian elimination to an augmented matrix, show that the average speed on each terrain type: sealed road, gravel road and snow/ice road is 200, 100 and 50 kilometres per hour respectively. Express your working in fractions to avoid rounding error during Gaussian elimination. [2 marks] (c) Suppose you prototype a car capable of an average speed of 150 km/h on sealed road, 100 km/h on gravel road and 80 km/h on snow/ice road. On which course i, orii) would it achieve the highest overall average speed [2 marks]?

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