Science:Math Exam Resources/Courses/MATH100/December 2011/Question 02 (b)/Solution 1

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

$\displaystyle {\frac {\ln(4/3)}{\ln(6/5)}}>1$

which means that the police arrived MORE than an hour after the murder, that is, the murder took place BEFORE 9:00PM.