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Please help ,calculate the maximum sustainable population

Please help ,calculate the maximum sustainable population-example-1

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Answer:


\sf P=\boxed{\sf 19.710}\;million

Explanation:

The population of New York state can be modeled by the function:


P(t) = (19.71)/(1+61.22e^(-0.03513t))

where:

  • P is the population in millions
  • t is the number of years since 1800.

The maximum sustainable population can be calculated by finding the limit of the population function P(t) as time (t) approaches infinity.

As t approaches infinity, the exponential term e^(-0.03513t) will approach zero. Therefore:


\begin{aligned}\displaystyle \lim_{{t \to \infty}} P(t) &=(19.71)/(1 + 61.22e^(-0.03513t))\\\\&= (19.71)/(1 + 61.22(0))\\\\&= (19.71)/(1 + 0)\\\\&= (19.71)/(1)\\\\&=19.71\end{aligned}

So, the limit of the population function as t approaches infinity is:


\displaystyle \lim_{{t \to \infty}} P(t)= 19.71

Therefore, the estimated maximum sustainable population of New York rounded to the nearest thousandth is:


\sf P=\boxed{\sf 19.710}\;million


\hrulefill

Note: When asked to round an answer to the nearest thousandth, we round to the third decimal place.

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