SOLUTION: The population of the US currently is growing with a doubling time of about 41 years. If the current population is 310 million of people, how many years will it take us to reach a

Algebra ->  Exponential-and-logarithmic-functions -> SOLUTION: The population of the US currently is growing with a doubling time of about 41 years. If the current population is 310 million of people, how many years will it take us to reach a      Log On


   

Question 768201: The population of the US currently is growing with a doubling time of about 41 years. If the current population is 310 million of people, how many years will it take us to reach a population of 350 million?
Years to target value (1 decimal):
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
For now,
Let +a+ = initial population
Let +P+ = current population
Let +k+ = years to double population
Let +n+ = number of +k+ doubling periods
it takes to reach target
-------------------------------
Initially,
+n+=+0+
+P+=+a%2A2%5En+
+P+=+a+
-------------------------
After +k+ years
+n+=+1+
+P+=+a%2A2%5E1+
-------------
After +2k+ years
+n+=+2+
+P+=+a%2A2%5E2+
--------------
So, in general, after +n%2Ak+ years,
+P+=+a%2A2%5En+
--------------
Now fill in values:
+k+=+41+ years
+a+=+310+ million
+P+=+350+ million
--------------
+350+=+310%2A2%5En+
+2%5En+=+350%2F310+
+2%5En+=+1.12903+
Take the log of both sides

+n+=+.05271+%2F+.30103+
+n+=+.1751+
+n%2Ak+=+.1751%2A41+
+n%2Ak+=+7.179+
In 7.2 years, the population of 350 million will be reached
--------------
check answer:
+350+=+310%2A2%5En+
+350+=+310%2A2%5E.1751+
+2%5E.1751+=+35%2F31+
+1.12904+=+1.12903+
close enough