 # Find a logistic function that describes the given population. Then Marenonigt 2021-12-29 Answered
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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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 $t=0\to {P}_{0}=300$
$300=\frac{5400}{1+Ax{e}^{0}}\to 1+A=\frac{5400}{300}=18$
$A=18-1=17$
$P\left(t\right)=\frac{5400}{1+\left(7\right){e}^{\mathrm{ln}\left(\frac{47}{119}t\right)}}$

We have step-by-step solutions for your answer! 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
$\frac{5400}{1+A}=300$
$1+A=18$
$A=17$

We have step-by-step solutions for your answer! Vasquez

We have step-by-step solutions for your answer!