Question on Natural Decay?

Question

There are no about 3300 different languages. These are derived from one original language, which develops into 1.5 languages every 6000 years. How long ago was the first original language spoken?
I don't understand how to start this problem!! please help

Answer 1

we get to use the exponential number and natural log for this one

B=Ae^(rt) A is the initial value, B is the resulting value after a time (t) and at a rate (r).
We know that after 6000 years we have 1.5 languages so
1.5=1e^(r*6000)
we don't know the rate for this but we can find it with this equation
ln(1.5)=6000r r=ln(1.5)/6000

now we have r so we can find the time after 3300 languages have been made

3300=1e^(rt)
ln(3300)=rt= ln(1.5)/6000 * t

6000ln(3300)/ln(1.5) = t
t is the time in years to create 3300 languages

Answer 2

I hope logarithms do not frighten you, because they are needed for this kind of problem.

We have 3300 languages, and each language becomes 1.5 over one period of 6000 years. So to go from 1 to 3300, we have to have 1.5 raised to the power of something, and that something is the number of 6000 years periods.

So we have:

3300 = 1.5 ^x

and we need to find x. Logs to the rescue, as this expression can be replaced by

log (3300) = log (1.5 ^x) = x * log (1.5)

3.5185 = x * 0.176

19.98= x

We might as well round this up to 20. So we had 20 cycles of 6000 years each, or 120 000 years.

To verify the result, lets look at the way it progresses.

At the beginning of the first cycle, there is 1 language.
After 6000 years, 1.5 (don't know what half a language means, but lets suppose this is a language that is half different and half the same)
After 12000 (2 cycles), we should have 1.5 * 1.5 = 2.25,
as the first original language has spawned a new language of its own, while the half language of before has turned into a full launguage and perhaps started to develop into dialects as well.

3 cycles: 1.5^3 or 3.375
4: 1.5^4 = 5.0625

and so on, until 20 cycles:
1.5 ^20 = 3325.25673.

Answer 3

The idea is that the number of languages grows exponentially. In 6000 years they go up by 50%.

The formula would be L = 1.5^(T/6000)

where L is the number of languages and T is years. (notice this works for 0, T = 1, and for 6000, T = 1.5)

So your question requires solving the equation

3300 = 1.5^(T/6000), which you can do with a calculator or with logarithms:

log(3300) = T/6000log(1.5)
T = 6000*log(3300)/log(1.5)

Answer 4

Hi. The words confuse, ignore them. The problem is this. We now have 3,300 lemons (say). 1.5 every 6,000 is the same as 2 every 8,000. Divide 3,300 by 2 and that is the number 8,000 year groups that you have until the origin. Make sense? (You can also do it by dividing 3,300 by 1.5 to reach the number of 6,000 year groups, but it's harder to understand.)