2. First show that for the completely insulated bar ut = c2uxx; 0 < x < L; t > 0: ux(0; t) = 0; t > 0 ux(L; t) = 0; t > 0 u(x; 0) = f(x); 0 < x < L the separation of variables gives the solution u(x; t) = A0 + 1 Ancos n x e ( cn )2t L n=1 L X where 2 Z0 L n x An = f(x)cos dx; n = 1; 2; 3:::; L L then nd the temperature in the bar with L = ; c = 1; and f(x) = x

Finally, the arena decides to offer advertising space on the jerseys of the arena’s own amateur volley ball team. the arena wants the probability of being shortlisted to be 0.14. what is this as a percentage and a fraction? what is the probability of not being shortlisted? give your answer as a decimal. those shortlisted are entered into a final game of chance. there are six balls in a bag (2 blue balls, 2 green balls and 2 golden balls). to win, a company needs to take out two golden balls. the first ball is not replaced. what is the probability of any company winning advertising space on their volley ball team jerseys?