1. ## Examquestion

Hi everybody, I have a problem with the following exercise, as I consequently get a wrong result, but it could also be that the solution is wrong.

1) The number of sms sent in Switzerland increases with every moth exponentially. In September 1999 21.5 Mio Sms were sent. In October it was 2.34 Mio Sms more.
In the following exercise I write in millions, so 10^6=1
a) How many SMS were sent in December 2000?
First, the ratio by which the number increases per month has to be found. This is done by calculating (21.5+2.34)/21.5 which equals around 1.109.

As there lie between september 1999 and december 2000 15 "steps", we can say 21.5*1.109^15= Number in December. As result I get 101.49 Millions, the solution is 110.(...) though.

b)When are 200 Mio SMS sent per month?

Here the formula is: 21.5*1.109^x=200
ln(200/21.5)/ln(1.109)=x=21.55...
Again, the solution is 20.5.. and not 21.5...

c) How many SMS were sent in the whole year 2000

We can consider the number of all sms sent in 2000 a geometric progression with a(1)= the number of sms in january 2000, the ratio r=1.109 and the number of repetitions n=11 (11 "steps" in year 2000)

a(1)= 21.5*1.109^4= 32.52...

so the Sum Formula for an GS is a(1-r^11)/(1-r)

I get as result 632.69..., the result in the booklet is 706.(..)

Can anybody help me out and say whether to booklet is wrong or where my mistake is?

Thanks a lot

2. I can't help for c), but both a) and b) appear to be correct. I treated it as a continous system, in which case, you have

$\frac{dP}{dt}=rP$

This gives that $P=Ae^{rt}$. Treating September 2009 as your base gives A=21.5. Taking a time unit of one month gives $21.5e^r=23.84$, giving my r to be 0.1033.

Then, after 15 time steps, $P=21.5e^{0.1033*15}=101.24$. I would assume the difference that we have between our answer is due to the number of decimal places or something similar.

Working in much the same way gives that the time step required to get 200 million SMS is 21.59 months. Again, I would ignore the minor difference between answers.

Your method is perfectly valid as well, just to note.