Science:Math Exam Resources/Courses/MATH100/December 2011/Question 02 (b)
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Question 02 (b) 

FullSolution Problems. In questions 28, justify your answers and show all your work. If a box is provided, write your final answer there. Simplification of answers is not required unless explicitly stated. A wealthy man was found murdered in his home. The police arrived on scene at 10:00 P.M. The temperature of the body at 10:00 P.M. was 33°C and one hour later it was 31°C. The temperature of the room in which the body was found was 21°C and normal body temperature is 37°C. Assume that the body cools after death according to Newton's Law of Cooling. Did the murder occur before 9:00 P.M.? Please justify your answer. 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! 
Hint 

Looking back to Question 2 (a), how long after the murder did the police arrive? When did the police arrive? Since you do not have access to a calculator during the exam, how can you evaluate ln(4/3)/ln(5/6) without a calculator? 
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. We know from Question 2 (a) that the police arrived ln(4/3)/ln(5/6) hours after the murder (we add a negative sign because we look at this as a positive amount of time now). So the question is whether ln(4/3)/ln(5/6) is larger than 1 or not. Let's see if this is meaningful or not. We know that ln(x) is positive if x is larger than 1. So ln(4/3) is positive and ln(5/6) is negative. Since ln(5/6) = ln(6/5), we can say that the (positive) amount of time it took for the police to get to the crime scene is ln(4/3)/ln(6/5). Which of 4/3 or 6/5 is larger? Well, they are the same as 20/15 and 18/15 respectively, so the denominator is smaller than the numerator and likewise if we take a logarithm, hence ln(4/3) > ln(6/5) and so which means that the police arrived MORE than an hour after the murder, that is, the murder took place BEFORE 9:00PM. 