Question

Please help ,calculate the maximum sustainable population

Answer 1

`\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.