SOLUTION: Solve the problem. Use the formula N = Iekt, where N is the population at time t, I is the initial population, and k is a growth constant equal to the percent of growth (expressed

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: Solve the problem. Use the formula N = Iekt, where N is the population at time t, I is the initial population, and k is a growth constant equal to the percent of growth (expressed       Log On


   

Question 600561: Solve the problem. Use the formula N = Iekt, where N is the population at time t, I is the initial population, and k is a growth constant equal to the percent of growth (expressed in decimal form) per unit of time. How long will it take for the population of a certain country to double if its annual growth rate is 7.7%? Round your answer to the nearest year.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
First off, let's use the fact that the population is growing at an annual rate of 7.7% to find the value of k.

If the initial population is say 1000 people, then in one year, we add 7.7% of that 1000 to the initial population to get: 1000 + 0.077*1000 = 1000 + 77 = 1077


So I = 1000 and N = 1077. This all happens in one year, so t = 1. Use these three variables to find k


N+=+Ie%5E%28kt%29

1077+=+1000%2Ae%5E%28k%2A1%29

1077+=+1000%2Ae%5E%28k%29

1077%2F1000+=+e%5E%28k%29

1.077+=+e%5E%28k%29

ln%281.077%29+=+ln%28e%5E%28k%29%29

ln%281.077%29+=+k%2Aln%28e%29

ln%281.077%29+=+k%2A%281%29

ln%281.077%29+=+k

0.0741793981742515+=+k

k+=+0.0741793981742515


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Now that you know the value of k, use it to find the value of t when I = 1000, N = 2000 (double the population of the initial 1000)


N+=+Ie%5E%28kt%29

2000+=+1000%2Ae%5E%280.0741793981742515t%29

2000%2F1000+=+e%5E%280.0741793981742515t%29

2+=+e%5E%280.0741793981742515t%29

ln%282%29+=+ln%28e%5E%280.0741793981742515t%29%29

ln%282%29+=+0.0741793981742515t%2Aln%28e%29

ln%282%29+=+0.0741793981742515t%2A%281%29

ln%282%29+=+0.0741793981742515t

ln%282%29%2F0.0741793981742515+=+t

0.693147180559945%2F0.0741793981742515+=+t

9.34420064896866+=+t

t+=+9.34420064896866

So it will take roughly 9.3442 years for the population to double.