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 Pre-Calculus and Calculus
 Exponential problem
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moon1130
Senior Member

USA
24 Posts

Posted - 09/10/2013 :  01:37:18  Show Profile  Reply with Quote
Problem:

"Suppose the population of a small country where no one moves away and everyone marries and stays married is 100,000. What will the population be in four generations if every couple has exactly two children?"

My Solution:

I am using the formula A=P(1+r)^t, where P is the beginning population of 100,000, t = the number of generations, and r =2 (doubling of the population per generation). Therefore,

A = 100,000(1+2)^4 = 8,100,000

Question:

I do not think that the rate of increase in population per generation is 200%. How do I determine the correct rate of change in this problem?

Thank yoiu
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moon1130
Senior Member

USA
24 Posts

Posted - 09/10/2013 :  07:56:12  Show Profile  Reply with Quote
I did the problem over again and tried to determine a value for r.
Now the population starts at 100,000. At the end of the 2nd generation we have a population of 100,000 + 200,00 = 300,000.
Therefore,
100,000(1+r)^2 = 300,000
(1+r)^2 = 3
1 + r = 3
r = 3 - 1 = 0.732..

Then A = 100,000(1+0.732)^4 = 900,000

So, after 4 generations, the population will be 900,000.

I would like to know if this solution is correct.

Thank you.
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moon1130
Senior Member

USA
24 Posts

Posted - 09/11/2013 :  14:31:21  Show Profile  Reply with Quote
My error. This is not an exponential problem but a geometric series problem.
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Ultraglide
Advanced Member

Canada
299 Posts

Posted - 09/23/2013 :  11:46:43  Show Profile  Reply with Quote
You're right, the increase is not 200% for each generation.
If we start with 100 000 people (all in couples) and each couple them have 2 children, then after 1 generation there will be 100 000 x 2 or 200 000 people. The population has obviously doubled. If it doubles in the next generation, there will be 400 000 (100 000 x 2). Can you continue the process?
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moon1130
Senior Member

USA
24 Posts

Posted - 09/26/2013 :  14:40:29  Show Profile  Reply with Quote
Thanks for your response, Ultraglide. After n generations, the population would be 100,000(2^n). Please let me know if I am incorrect.

Again, thank you very much.
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Ultraglide
Advanced Member

Canada
299 Posts

Posted - 09/26/2013 :  19:02:44  Show Profile  Reply with Quote
Yes.
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