13786 10 In a factory, a biological substance is placed in a large tank. During the production process, the biological substance reproduces at a rate of 0.25A kg per hour, where A kg is the amount of substance present at time t hours. At the same time, the biological substance is pumped out of the tank at a constant rate of 50 kg per hour. (i) By setting up and solving a suitable differential equation, show that A = 4ke0.25t + 200 [7] where k is a constant. 13786 [Turn over At time t = 0, A = 190 (ii) Show that A = 200 – 10e0.25t [2] 13786 The tank must be completely empty by the end of each production shift. (iii) Find the minimum length of time for each shift. Note that a production shift always lasts for a whole number of hours. [3] 13786 The factory management decides to change to 8-hour production shifts. They plan to manage this by simply reducing the amount of biological substance in the tank at time t = 0 (iv) Find the maximum amount of biological substance at time t = 0 which makes this change feasible, giving your answer to 1 decimal place. [5] Permission to reproduce all copyright material has been applied for. In some cases, efforts to contact copyright holders may have been unsuccessful and CCEA will be happy to rectify any omissions of acknowledgement in future if notified. Examiner Number Sources: All images © CCEA DO NOT WRITE ON THIS PAGE 13786/6 For Examiner’s use only Question Number Marks 1 2 3 4 5 6 7 8 9 10 Total Marks THIS IS THE END OF THE QUESTION PAPER