# Math Help - Population Question

1. ## Population Question

A population, $P$, is said to be growing logistically if the time, $T$, taken for it to increase from $P_1$ and $P_2$ is given by:

$T=\int_{P_1}^{P_2} \frac {kdP}{P(L-P)}$

where $k$ and $L$ are positive constants and $P_1 < P_2 < L$.

A) Calculate the time taken for the population to grow from $P_1 = \frac {L}{4}$ to $P_2 = \frac {L}{2}$.

B)What happens to $T$ as $P_2$ approaches $L$?

I honestly have no clue how to even start this problem. Any help would be appreciated.

Thanks!

2. Originally Posted by tiace
A population, $P$, is said to be growing logistically if the time, $T$, taken for it to increase from $P_1$ and $P_2$ is given by:

$\int_{P_1}^{P_2} \frac {kdP}{P(L-P)}$

where $k$ and $L$ are positive constants and $P_1 < P_2 < L$.

A) Calculate the time taken for the population to grow from $P_1 = \frac {L}{4}$ to $P_2 = \frac {L}{2}$.

B)What happens to $T$ as $P_2$ approaches $L$?

I honestly have no clue how to even start this problem. Any help would be appreciated.

Thanks!
To do the integration you'll need to use Partial Fractions. Then substitute $P_1 = \frac{L}{4}$ and $P_2 = \frac{L}{2}$ as your terminals.

3. Is this integral correct?

$T=\int_{P_1}^{P_2} \frac {kdP}{P(L-P)} = \frac {-k}{L} * (\ln|P-L|-\ln|P|)$

I'm not sure where to go from here though.

4. Yes that's fine.

$T=(\frac {-k}{L} * (\ln|\frac{L}{2}-L|-\ln|\frac{L}{2}|))-(\frac {-k}{L} * (\ln|\frac{L}{4}-L|-\ln|\frac{L}{4}|))$
$T=\frac{k\ln{3}}{L}$