# Thread: half life, growth rates

1. ## half life, growth rates

A population doubles every 15 years. Determine the yearly growth rate

A radioactive substance decays from 400 grams to 120 grams in 320 years. Determine it's half life

Hey guys, as far as i know the above problems should be solvable with the formula A=I*r^(t/p)

A being the final value
I being the initial value
r being the rate of growth/decay
t being the time
and p being the growth/decay period per rate

It would be great if anyone can help. Thanks in advance

2. Originally Posted by imppy725
A population doubles every 15 years. Determine the yearly growth rate

A radioactive substance decays from 400 grams to 120 grams in 320 years. Determine it's half life

Hey guys, as far as i know the above problems should be solvable with the formula A=I*r^(t/p)

A being the final value
I being the initial value
r being the rate of growth/decay
t being the time
and p being the growth/decay period per rate

It would be great if anyone can help. Thanks in advance
I believe the equation you should be using is:
A = I*e^(rt)

Where e is the natural number: e = 2.71828182846...

For the first problem, we know that A = 2I when t = 15:
2I = I*e^(15r)
2 = e^(15r)
ln2 = lne^(15r)
ln2 = 15r
r = ln2/15 = 0.0462

For the second problem, we know that I = 400, A = 120 when t = 320:
120 = 400*e^(320r)
1/3 = e^(320r)
ln(1/3) = 320r
r = ln(1/3)/320

Now, we want A = 1/2(I) = 1/2(400) = 200:
200 = 400*e^(t*ln(1/3)/320)
1/2 = e^(t*ln(1/3)/320)
ln(1/2) = t*ln(1/3)/320
t = 320*ln(1/2)/ln(1/3) = 201.8975