Exponential Growth/Decay Calculus Problem?

Question

If a bateria culture starts with 8000 bateria and doubles every 30 minutes, how many minutes will it take the population to reach 38000?

Any help would be great!

Answer 1

N(t) = 8000 . 2^(t/30)
So 38000 = 8000 . 2^(t/30)
=> 2^(t/30) = 38000/8000 = 4.75
=> (t/30) log 2 = log 4.75
=> t = 30 log 4.75 / log 2 = 67.4 minutes (1 d.p.)

Answer 2

We can write
P/Po = exp(k t), where k is a growth constant in minutes-1, t is time, Po is your initial bacteria and P is your bacteria at any time t in the exponential growth phase (this can't go on forever). exp is the natural log base "e", or 2.718
You can convert your double time to k
0.693 = 30 k, k = 0.0231/min
You know P/Po= 4.75, so
4.75= exp (0.0231 t) and solve for t.

Answer 3

Take y=ce^kt
y=8000e^kt-------equation 1
then
2(8000)=8000e^k(1/2)
2=e^1/2k
therefore k=1.38629,sub back to the equation 1
38000=8000e^1.38629t
4.75=e^1.38629t
taking In at both side
In 4.75=1.38629t
t=1.1236hr
convert to mins
then x60min

The answer will be 67.416 mins