MHB How Does Exponential Growth Affect a Bacterial Population?

Tineeyyy
Messages
2
Reaction score
0
A bacterial population x is known to have a growth rate proportional to x itself. If between 12 noon and 2:00 pm, the popilation triples.
1. What is the growth rate pf the given problem?
2. Is the number of bacterial population important to the problem?
3. At what time should x become 100 times what it was at 12 noon assuming that no controls are being exerted
 
Physics news on Phys.org
Tineeyyy said:
A bacterial population x is known to have a growth rate proportional to x itself.
So dx/dt= kx for some k.
dx/x= kdt. Integrating, ln(x)= kt+ c
Taking the exponential, $x= e^{kt+ c}= Ce^{kt}$ where $C= e^c$.

If between 12 noon and 2:00 pm, the popilation triples.
Taking t to be the time in hours since noon, $X(0)= C$, $X(2)= Ce^{2k]= 3C$, e^{2k}= 3$.

1. What is the growth rate pf the given problem?
$e^{2k}= 3$ so $2k= ln(3)$ and $k= \frac{1}{2}ln(3)$.

2. Is the number of bacterial population important to the problem?
That's a strange question! Obviously the number is the whole point of the problem! But I suspect they mean the initial number of bacterial population, X(0). No, that cancels so is not at all important.

3. At what time should x become 100 times what it was at 12 noon assuming that no controls are being exerted.
We want to solve X(t)= 100X(0).
$X(t)= X(0)e^{kt}$ and we have determined that $k= \frac{1}{2}ln(3)= ln(\sqrt{3})$ so $X(t)= X(0)e^{ln(\sqrt{3})t}= 100X(0)$.

$e^{ln(\sqrt{3}t}= \sqrt{3}^t= 100$.
$t ln(\sqrt{3})= t\left(\frac{1}{2} ln(3)\right)= 100$.
$t= \frac{200}{ln(3)}$

So about 182 hours or 7 days 14 hours.
 
For original Zeta function, ζ(s)=1+1/2^s+1/3^s+1/4^s+... =1+e^(-slog2)+e^(-slog3)+e^(-slog4)+... , Re(s)>1 Riemann extended the Zeta function to the region where s≠1 using analytical extension. New Zeta function is in the form of contour integration, which appears simple but is actually more inconvenient to analyze than the original Zeta function. The original Zeta function already contains all the information about the distribution of prime numbers. So we only handle with original Zeta...
Back
Top