# Find a logistic function that describes the given population. Then

Find a logistic function that describes the given population. Then graph the population function. The population increases from 300 to 700 in the first year and eventually levels off at 5400. Write the equation of a logistic function that models the given population. $P\left(t\right)=$ (Type an exact answer.)
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Bob Huerta
Population increases 300 to 700
levels off at 5400.
Logistic equation is given by
$P\left(t\right)=\frac{L}{1+A×{e}^{B\left(t\right)}}$
$B=\frac{1}{{f}_{1}}×\mathrm{ln}\left(\frac{{P}_{0}\left({P}_{5}-{P}_{1}\right)}{{P}_{1}\left({P}_{5}-{P}_{0}\right)}\right)\right]$
${P}_{0}\to \text{initial population}$
${P}_{1}\to \text{after}\left({f}_{1}\right)\text{population}$
${P}_{5}\to \text{level of population}$
$B=\mathrm{ln}\left[\frac{300\left(5400-700\right)}{700\left(5400-300\right)}\right]$
$B=\mathrm{ln}\left[\frac{300×4700}{700×5100}\right]=\mathrm{ln}\left[\frac{47}{119}\right]$
at $t=0\to {P}_{0}=300$
$300=\frac{5400}{1+Ax{e}^{0}}\to 1+A=\frac{5400}{300}=18$
$A=18-1=17$
$P\left(t\right)=\frac{5400}{1+\left(7\right){e}^{\mathrm{ln}\left(\frac{47}{119}t\right)}}$

abonirali59

The logistic equation has general form
$f\left(x\right)=\frac{L}{1+A{e}^{Bx}}$ (1)
$L=5400$, here x is to find A and B
${P}_{0}=\text{initial population}$
${P}_{1}=\text{after}{t}_{1}\text{population}$
${P}_{5}=\text{level of population}$
from equation (1)
$B=\mathrm{ln}\left[\frac{300\left(5400-700\right)}{700\left(5400-300\right)}\right]=\mathrm{ln}\left[\frac{300×4700}{700×5100}\right]$
$B=\mathrm{ln}\left[\frac{47}{119}\right]$
After getting a put t, to find A
$\frac{5400}{1+A}=300$
$1+A=18$
$A=17$

Vasquez