The population growth of a certain rodent is approximately 3% per month.?

Find the time it takes for the population to triple.





Thank You!!!The population growth of a certain rodent is approximately 3% per month.?
in general about 37 months





The thing to realize is that, to find the population in the next month, we multiply the previous month population by 1.03. For example, if we started with 100, in the next month we have 100+100*3% = 100 + 100*0.03 = 100*(1+0.03) = 100*1.03. That amounts to 103. In the month after that, we would do the exact same thing, and end up muliplying by 1.03 to get to the next month. So, if we start with x rodents, after n months we have,





x*1.03*1.03*1.03*......n times


=x*(1.03)^n





We want the population to triple, or to reach 3*x (3 times what you started with). Thus, we want,





x*(1.03)^n = 3*x





Solve it...





x*(1.03)^n - 3*x = 0


x * [(1.03)^n - 3] = 0





So, we have two options. If x =0, then we win right away. We start with zero rodents from day 1, and 0*3 = 0, so we have already reached 3 times as many rodents!!!





The other option is that





(1.03)^n - 3 = 0


(1.03)^n = 3





Take ln of both sides...


n*ln1.03 = ln3


n = ln3/ln(1.03)


or


n = about 37 months.





So after about 37 months it triples.The population growth of a certain rodent is approximately 3% per month.?
37.17 months





P(t) =Po (1.03)^t , where t is in months and P is population


exponential growth with a constant growth rate of 3% and initial population is Po





3 = (1.03)^t for tripling


ln 3 = t ln 1.03


t = ln 3 / ln 1.03 = 37.17 months

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