Logistic growth: P(t)=(C)/(1+a e^-k t) For populations that exhibit logistic growth, the populat

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Logistic growth: P(t)=(C)/(1+a e^-k t) For populations that...

Logistic growth: $P(t)=\frac{C}{1+a e^{-k t}}$ For populations that exhibit logistic growth, the population at time $t$ is modeled by the function shown, where $C$ is the carrying capacity of the population (the maximum population that can be supported over a long period of time), $k$ is the growth constant, and $a=\frac{c-P(0)}{P(0)} .$ Solve the formula for $t$, then use the result to find the value of $t$ given $C=450, a=8, P=400,$ and $k=0.075$.

Logistic growth: $P(t)=\frac{C}{1+a e^{-k t}}$
For populations that exhibit logistic growth, the population at time $t$ is modeled by the function shown, where $C$ is the carrying capacity of the population (the maximum population that can be supported over a long period of time), $k$ is the growth constant, and $a=\frac{c-P(0)}{P(0)} .$ Solve the formula for $t$, then use the result to find the value of $t$ given $C=450, a=8, P=400,$ and $k=0.075$.

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Key Concepts

Algebraic Manipulation

Solving for the time variable in a logistic growth model requires rearranging the equation and applying algebraic operations. This process involves isolating the exponential term and then applying the logarithm to both sides to solve for the variable of interest.

Logarithms

Logarithms are the inverse of exponential functions and are used to solve equations where the unknown variable is in an exponent. In logistic growth equations, logarithms allow one to isolate the time variable by 'undoing' the exponential function.

Exponential Function

An exponential function appears in logistic growth models to represent the dynamic phase of growth. Its presence in the denominator of the logistic function accounts for the declining growth rate as the population nears carrying capacity.

Carrying Capacity

The carrying capacity is the upper limit of the population size that the environment can sustain indefinitely. It determines the maximum value that the logistic growth function can approach as time goes on.

Logistic Growth Model

This model represents population growth that is initially exponential, but slows as the population approaches a maximum limit. It is characterized by a rapid increase in size that eventually levels off due to environmental resistance and other limiting factors.

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