how does this work? population P=20-(4/(t+1)), t=0 is 1990; when is P=19K?

[allegansveritatem]

Here is the problem:



Here is my solution:



After I worked this out I got a little suspicious of this formula because it seemed only to work as long as the population didn't get larger than 19000. When I tried to use it to determine how long it would be before the population got to 40000, here is what I got:



Am I handling this wrong or what?

 

[Dr.Peterson]

Here is the problem:

View attachment 10462

Here is my solution:

View attachment 10463

After I worked this out I got a little suspicious of this formula because it seemed only to work as long as the population didn't get larger than 19000. When I tried to use it to determine how long it would be before the population got to 40000, here is what I got:

View attachment 10464

Am I handling this wrong or what?

No, you have just discovered that the population will never reach 40,000; in fact, it will never be above 20,000.

If you have learned anything about rational functions, you might think about asymptotes. Or you could use a graphing utility (e.g. desmos.com) and take a look at how P varies.

Or, just notice that you are subtracting from 20 a quantity that will always be positive when t is positive. So P is always less than 20.

Also, think about what the population was in 1989 (i.e. t=-1)!

 

[allegansveritatem]

No, you have just discovered that the population will never reach 40,000; in fact, it will never be above 20,000.

If you have learned anything about rational functions, you might think about asymptotes. Or you could use a graphing utility (e.g. desmos.com) and take a look at how P varies.

Or, just notice that you are subtracting from 20 a quantity that will always be positive when t is positive. So P is always less than 20.

Also, think about what the population was in 1989 (i.e. t=-1)!

I did notice that, at least judging from this formula, anything above 19000 is not going to happen....Seems to be this formula is defective in that it can only serve to model the population growth within a very narrow range. But maybe I am missing something that the writers of the formula knew, say, that the land within the city precincts was such that it could not support a greater population than 19000.

 

[Dr.Peterson]

I did notice that, at least judging from this formula, anything above 19000 is not going to happen....Seems to be this formula is defective in that it can only serve to model the population growth within a very narrow range. But maybe I am missing something that the writers of the formula knew, say, that the land within the city precincts was such that it could not support a greater population than 19000.

I don't think they're claiming to be realistic. It's just an exercise. They probably just wanted to give you a rational equation to solve, rather than wanting to teach you how to estimate real-life population growth.

Any population below 20,000 will occur (though not all at whole numbers of years). Where did you get 19,000 as a maximum? How do you conclude that, say, 19,500 will not happen?

There are, of course, situations where population (say, of fruit flies in a container) has an upper limit; and there are much more appropriate models (such as logistic growth) that are used in those cases. You'll probably see those when you learn about logarithms, and again in calculus or differential equations. But even then, it will be an idealized formula. In classroom exercises, the data will be fake, just to give a chance to practice the algebra.

 

[allegansveritatem]

I don't think they're claiming to be realistic. It's just an exercise. They probably just wanted to give you a rational equation to solve, rather than wanting to teach you how to estimate real-life population growth.

Any population below 20,000 will occur (though not all at whole numbers of years). Where did you get 19,000 as a maximum? How do you conclude that, say, 19,500 will not happen?

There are, of course, situations where population (say, of fruit flies in a container) has an upper limit; and there are much more appropriate models (such as logistic growth) that are used in those cases. You'll probably see those when you learn about logarithms, and again in calculus or differential equations. But even then, it will be an idealized formula. In classroom exercises, the data will be fake, just to give a chance to practice the algebra.

well, it could go a little above 19000. I was just using that as a nice round number., I haven't gotten to logarithms, at least not this time around. Seems to me long long ago I knew something about them but in those days I had too much on my mind to really learn much of anything. Math is like music, you have to practice it. And practice. And practice.

 

[Steven G]

well, it could go a little above 19000. I was just using that as a nice round number.

Actually it can go much over 19,000 (actually it can get close to 20,000. If you let t be very large, then 4/(t+1) will be quite small. Then when you subtract this value from 20,000 you'll get close to 20,000.

 

[Dr.Peterson]

well, it could go a little above 19000. I was just using that as a nice round number.

I thought 20,000 was a nice round number.

The population can be anything below that. It just can't reach exactly 20,000. Actually, come to think of it, the actual population has to be an integer, and eventually, the formula will round to 20,000!