Exponential growth & decay

Question

the population of a city grows at a rate of 5% per year. The population in 2000 was 400,00. What would be the predicted current populatoin? in what year would we predict the population to reach 1,000,000?

Answer 1

 

P(t) =  N*(1.05)^(t-2000) where N is the starting population and t is the Year

 

Note that 5% growth means the population increases 100%+5% = 105% = 1.05

 

and year 2000 be the starting year, which is why 2000 is subtracted in the expoenent

 

P(2000) = 400000

N*(1.05)^(2000-2000) = 400000

N*(1.05)^0 = 400000

N*1 = 400000

N = 400000

 

The population function is

P(t) = 400000*(1.05)^(t-2000)

 

 

P(2018) = 400000*(1.05)^(18) = 962647.69347.....

 

1000000 = 400000(1.05)^(T-2000)

 

1000000/400000 = (1.05)^(t-2000)

 

2.5 = 1.05^(t-2000)

 

LOG 2.5 = LOG 1.05 ^ (t-2000)

 

log 2.5 = (t-2000) LOG 1.05

 

log 2.5 / log 1.05 = t - 2000

 

T = Log 2.5 / Log 1.05 + 2000

 

  = 2018.78023456328.....

 

12 months per year * 0.78023456... = 9.37

So in September this year the population will hit 1 million