# The population of a culture of bacteria is modeled by the logistic equation P (t) = frac{14,250}{1 + 29e – 0.62t} To the nearest tenth, how many days will it take the culture to reach 75% of it’s carrying capacity? What is the carrying capacity? What are the virtues of logistic model?

The population of a culture of bacteria is modeled by the logistic equation $P\left(t\right)=\frac{14,250}{1+29e–0.62t}$ To the nearest tenth, how many days will it take the culture to reach 75% of it’s carrying capacity? What is the carrying capacity? What are the virtues of logistic model?
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Step1 $P\left(t\right)=\frac{14,250}{1+29{e}^{\left(}-0\ast 62t\right)}$
$Ast⇒\mathrm{\infty },{e}^{-0\ast 62t}⇒0$
$\therefore P\left(t\right)⇒14,250$.

This is the carrying capacity of the bacten`al culture.

To determine the time it take to reach 75% of thr carlying capacity, we write $P\left(t\right)=\frac{3}{4}\left(14,250\right).$

Step 2

$⇒\text{⧸}14,2501+29{e}^{\left(-0\ast 62t\right)}=\frac{3}{4}\left(\text{⧸}14,250\right)$
$⇒1+29{e}^{-0\ast 62t}=\frac{3}{4}$
$⇒{e}^{\left(-0\ast 62t\right)}=\frac{1}{87}$
$⇒t=\frac{1}{0\ast 62}In\left(87\right)=7\cdot 2$ Hense it takes the bacterial Popylation $7\cdot 2$ days to reach 75% of the calling capacity.

Step 3 The growth of any population is curtailed by the amount of resources available. This is better modeled by a logistic S curve rather than an exponential growth model which only works in ideal situations and not for modeling real life situations.