The relationship between the elapsed time t in years since t

The relationship between the elapsed time, t, in years, since the closing of the mill, and the town\'s population. P (t), is modeled by the following function. P (t) = 12,000 - 2^-1/15

Solution

We have P(t) = 12000*2^-t/15 where P(t) is the population of Sawyervillie t years after the mill closed down. When P(t) = 9000, we have 9000 = 12000*2^-t/15 or, 2^-t/15 = 9000/12000 = 3/4. Now, on taking logarithm of both the sides, we get log(2^-t/15) = log(3/4) or, (-t/15) log 2 = (log 0.75) or, t = -15(log 0.75)/log2 = -15(-0.124938736/0.301029995 = 6.225562473 years or, 6.23 years( on rounding off to the nearest hundredth). The answer is 6.23 years.