MHB What is the best deal for maximum profit?

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The discussion focuses on the mathematical model P(t) = (500)/[1 + 82.3e^(-0.162t)], which describes the growth of an endangered species population. Participants confirm that as time (t) approaches infinity, the limit of P(t) is indeed 500, as the exponential term approaches zero. The graph of P(t) can be used to visualize this limit, with points plotted in the form (t, P(t)). The conversation also touches on the nature of the function, noting it is exponentially decaying due to the negative exponent. Ultimately, the best deal for maximum profit is implied to be the one that maximizes this population growth.
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The model below is given to find the growth of a population of an endangered species.

P(t) = (500)/[1 + 82.3e^(-0.162t)]

Find the limit of P(t) as t tends to positive infinity.

The answer in the textbook is 500.

Can a model like this be graphed? If so, is the graph of P(t) the best approach to find the limit?
Points on the graph of P(t) are in the form (t, P(t)). Correct?
 
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Beer soaked ramblings follow.
nycmathdad said:
The model below is given to find the growth of a population of an endangered species.

P(t) = (500)/[1 + 82.3e^(-0.162t)]

Find the limit of P(t) as t tends to positive infinity.

The answer in the textbook is 500.

Can a model like this be graphed? If so, is the graph of P(t) the best approach to find the limit?
Points on the graph of P(t) are in the form (t, P(t)). Correct?
Problem 1.5.79.a.
Again, you left out a lot of details.
A screenshot will be helpful.
The problem is asking you to find $\lim_{t \to \infty} P(t)$
 
I am interested in 79 partn(a) only.

Here it is:

Screenshot_20210402-185104_Drive.jpg
 
nycmathdad said:
The model below is given to find the growth of a population of an endangered species.

P(t) = (500)/[1 + 82.3e^(-0.162t)]

Find the limit of P(t) as t tends to positive infinity.

The answer in the textbook is 500.

Can a model like this be graphed? If so, is the graph of P(t) the best approach to find the limit?
Points on the graph of P(t) are in the form (t, P(t)). Correct?

It's an exponentially decaying function due to the negative power on e. So what value do you think that the exponential function tends to?
 
Prove It said:
It's an exponentially decaying function due to the negative power on e. So what value do you think that the exponential function tends to?

Well, 82.3e^(-0.162t)] becomes zero leaving the expressing 500/1. The limit is 500.
 
nycmathdad said:
Well, 82.3e^(-0.162t)] becomes zero leaving the expressing 500/1. The limit is 500.

Correct.
 
Prove It said:
Correct.

Me correct? Really? Wow!
 
Beer soaked non sequitur ramblings follow.
nycmathdad said:
Me ...
The best deal is the one that brings the most profit.
 

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