1 Questions Paper for BITI 1223 Calculus & Numerical Methods March 23, 2016 Instructions to the Students 1. Write your MATRIX NUMBER, NAME, and TUTORIAL GROUP at the top-right corner of the first page of your answer sheets. 2. There are 5 (FIVE) questions. Answer ALL questions. 3. Show all necessary workings neatly. 4. You may use pens to write the answer for questions ; but your handwriting must be readable. 5. Your answers have to be submitted by 22 April 2016, 5 pm. at your lecturer's office. Questions 1. Question 1: 1 of 10 Marks The number of bacteria N in a culture is given by the model N = 240ekt where t is time in hours and t = 0 corresponds to the time when N = 240. When t = 10, the bacteria count is N = 320. How long does it take for the population of bacteria to reach N = 720 ? 2. Question 2: 1 of 10 marks Find the volume of the largest open box that can be made from a piece of cardboard 24 inches square by cutting equal squares from the corners and turning up the sides 3. Question 3: 2 of 10 marks A cannonball is dropped from a height of 179 feet above the ground. The height of the ball after t seconds is given by s(t) = 179 16t 2 . Find the velocity and acceleration functions. Compute the instantaneous velocity of the cannonball at 2 seconds and the acceleration of the ball at 2 seconds 4. Question 4: 3 out of 10 marks A model for the spread of a rumour is that the rate of spread is proportional to the product of the fraction of the population who have heard the rumour and the fraction who have not heard the rumour. The differential equation that models this is dy = ry(1 y) dt Where y denotes the fraction of the population who has heard the rumour and r denotes the rate at which the rumour spread. The solution of this differential equation is, y= y0 y0 + (1 y0 )ert (Be sure to show all the details when you solve your differential equation) Then, assume a small town has 2000 peoples. At 8 AM, 80 people have heard a rumour. By noon, half the town has heard it. We would like to know at what time 90% of the population have heard the rumour. 1 5. Question 5: 3 out 10 marks A certain radioactive material is known to decay at a rate proportional to the amount present. If initially there is 50 milligrams of the material present and after two hours it is observed that the material has lost 10 percent of its original mass, find (a) an expression for the mass of the material remaining at any time t, (b) the mass of the material after four hours, (c) the time at which the material has decayed to one half of its initial mass. 2