$\text { The logistic function } P ( t ) = \frac { 10 } { 1 + 3 e ^ { - 2 t } } \text { models the growth of a population. }$ Determine when the population reaches one-half of the maximum carrying capacity. Round your answer to three decimal places.
A)0.549
B)3.333
C)1.151
D)5.000
E)1.000

Question

Quizplus · Accepted Answer

The logistic function is given by $P ( t ) = \frac { L } { 1 + b e ^ { - k t } } $, where $L$ is the maximum carrying capacity. In this case, $L = 10$. We want to find when $P ( t ) = \frac { L } { 2 } = 5$. Substituting this into the logistic function, we get:$$5 = \frac { 10 } { 1 + 3 e ^ { - 2 t } }$$Solving for $t$, we get:$$t = 0.549$$Therefore, the population reaches one-half of the maximum carrying capacity at $t = 0.549$.

The logistic function P(t)=101+3e−2t models the growth of a population. \text { The logistic function } P ( t ) = \frac { 10 } { 1 + 3 e ^ { - 2 t } } \text { models the growth of a population. } The logistic function P(t)=1+3e−2t10​ models the growth of a population.

$\text { The logistic function } P ( t ) = \frac { 10 } { 1 + 3 e ^ { - 2 t } } \text { models the growth of a population. }$