$\text { The logistic function } P ( t ) = \frac { 15 } { 1 + 5 e ^ { - 4 t } } \text { models the growth of a population. }$ Determine when the population reaches of the maximum carrying capacity. Round your answer to three decimal places.
A)4.317
B)3.000
C)0.474
D)0.677
E)0.301

Question

Quizplus · Accepted Answer

The logistic function $P(t) = \frac{15}{1 + 5e^{-4t}}$ models population growth with a maximum carrying capacity of 15, as indicated by the numerator. The population reaches its maximum carrying capacity asymptotically as $t$ approaches infinity, not at a specific finite time. Therefore, none of the given options directly indicate when the population reaches its maximum carrying capacity. However, the question seems to be misinterpreted. If it's asking for when the population growth rate is highest, which is at the inflection point, we solve the derivative of $P(t)$ for its maximum growth rate. The inflection point occurs when $t = \frac{\ln(5)}{4}$, which is approximately 0.401, not listed among the options. Given the options, the closest to any relevant calculation would be E) 0.301, but this is based on a misunderstanding of the question as presented.

The logistic function P(t)=151+5e−4t models the growth of a population. \text { The logistic function } P ( t ) = \frac { 15 } { 1 + 5 e ^ { - 4 t } } \text { models the growth of a population. } The logistic function P(t)=1+5e−4t15​ models the growth of a population.

$\text { The logistic function } P ( t ) = \frac { 15 } { 1 + 5 e ^ { - 4 t } } \text { models the growth of a population. }$