In just 5 weeks the population goes from 100 to 1000. What is the doubling time of this population? As in, how long does the population take to double?

i tried 1000=100(2)(t/5), where t is time in weeks

10=2(t/5)

log210=t/5

log(10)/log(2)=t/5

5*3.322=t

t=16.61

16.6 is quite a bit more than 5, which was the number of weeks it took to grow by 10x, so I knew that obviously wasn't correct. Can anyone help me figure out how this is solved or what formula is used?

Aleguan
Feb 1, 2018

#1**+1 **

We have this function :

P(t) = 100(e)^{kt}

And when t = 5, we have that

1000 = 100(e)^{k5} divide both sides by 100

10 = e^{5k} take the ln of both sides

ln 10 = ln e^{5k} and we can write

ln 10 = 5k ln e { ln e = 1...so we can ignore this }

ln 10 = 5k divide both sides by 5

ln 10 / 5 = k

The function is

P(t) = 100(e)^{ ln10 * t / 5}

So.....we want to find the doubling time....and we have that

200 = 100(e)^{ln10 * t / 5} divide both sides by 100

2 = e^{ ln 10 * t / 5} take the ln again

ln 2 = ln e ^{ln 10 * / 5} amd we can write

ln 2 = [ (ln10) / 5 ] * t

5 ln 2 = ln 10 * t

5 ln 2 / ln 10 = t ≈ 1.5 weeks

CPhill
Feb 1, 2018