Engineering Mathematics 2

End of Term Worksheet Assessment Note: Write your Student Number at the top of all pages submitted. Show all working in the derivations you make and spread your working out well. I suggest you use A4 sized, lined paper to help with the layout and that you staple the pages. Hand your solutions in at the School Office, at the latest by 4:00 pm on Thursday 10th December. Attempt all FIVE questions. 1. (a) For the first order initial value problem: , ?0? 5, 2 ? xy y ? dx dy use the Euler iteration method i.e. n n hfn y ?1 ? y ? to calculate the values yn+1, for n = 0,1,2 using a step size h = 0.1. [3 marks] (b) By separating variables solve the equation exactly and hence compute the errors in the Euler approximation for the three iteration steps (writing your answers to 6 decimal places). [5 marks] 2. Consider a bar of metal of unit length in which heat transfer between the bar and it surroundings is assumed to obey the heat equation given by: , 0 1, 0 2 2 ? ? ? ? ? ? ? ? ? u x t x u t u u?0,t? ? u?1,t? ? 0, t ? 0 u?x,0? ? f ?x?, 0 ? x ?1. (a) By assuming that the solution is of the form: u?x,t? ? X?x?T?t?, show that: X?? ? ?X ? 0 and T? ? ?1? ??T ? 0 where ? is an unknown constant. [4 marks] (b) By considering the X variable equation for all possible ? and applying appropriate boundary conditions, show that the only possible (non-trivial) solutions are: X ? A sin?n x?, n ?1,2,3,... n n ? corresponding to the choices: , 1,2,3,... 2 2 ?n ? ?n ? n ? where An are unknown constants. [10 marks] (c) Hence, obtain the solution: ? ? ? ? n t n n t u x t e C n x e 2 2 , sin 1 ? ? ? ? ? ? ? ? where the Cn are left as unknown constants. [4 marks] 3. It is estimated that the probability of a rocket exploding during lift-off is 0.02 and that the chance of its guidance system failing is 0.05, where these events are independent. Find the probabilities that: (a) The rocket will not explode during lift-off. [2 marks] (b) It will either explode or have its guidance system fail. [2 marks] (c) It will not explode and not have a guidance system failure. [2 marks] 4. At a service till customers arrive at an average rate of 2.5 per minute. (a) Write down the statistical model for the number of people arriving each minute. [1 mark] (b) Find the probability that at most 3 will arrive in any given minute. [2 marks] (c) Find the probability that at least 3 will arrive during an interval of two minutes. [3 marks] (d) Use the Normal approximation to find the probability that at least 20 will arrive in an interval of six minutes. [3 marks] 5. The mileage (in thousands of miles) that car owners get from a certain type of tyre is a random variable x having the probability density: f (x) = 0 for x < 0 ? ? 20 x 20 e f x ? ? for x ? 0. (a) Find the mean mileage achieved with this type of tyre. [3 marks] (b) Find the probability that a tyre fails before 10,000 miles have been driven on it. [2 marks] (c) Find the probability that it lasts for at least 30,000 miles. [2 marks] (d) Find the median lifetime of the tyre. [2 marks]