The Decline Of Justin Bieber

The classic declining exponential function is of the form a e^(-bx). Here, we let a = 5000 and x = m-1, where m is the month number, so that F(m) = 5000 e^(-b(m-1)). We have F(1) = 5000 and F(7) = 4000, from which we determine b to be (1/6)Log(5/4). From this, we see that F(25) = 2048 and F(26) = 1973.23. January 2015 comes closest to this.

In 6 months, it declined by $\frac{4000}{5000} = 0.8$ . We want to know how long it takes to decline to $\frac{2000}{5000} = 0.4$ .

Since $0.8^4 = 0.4096 \approx 0.4$ , hence it would take us close to $4 \times 6$ months, which would be January 2015 .

It says that the rate of decline is proportional to the number of radio stations. So, rate = kN where k is the rate constant. rate is the number of stations per unit time and can be transformed into dN/dt.
dN/dt = kN -> dN/N = kdt

integrating dN/N from 5000 to 4000 and dt with 1 to 7(Jan. to July) gives a value of k = -0.0372

integrating dN/N from 5000 to 2000 and dt with 1 to x(Jan. to required time) gives a value of x = 25.6 in months

25.6/12 = 2.13 years so, 2013 + 2 = 2015. 25.6 - 24 = 1 month so, the answer is Jan 2015.

I think the answer can be APPROXIMATED by using geometric progression an=a1r^(n-1). Since a1=5000 and a7=4000 (july is the seventh month), the common ratio (rate of decline) is (4/5)^(1/6). Using the formula for GP and by letting an=2000, we can solve for n. n=6[log(2/5)/log(4/5)]+1= 25.64 (inclusive of january 2013). January 2013+ 25 months= Feb 2015. I picked January 2015 because its the closest.