# Rate of increase of population of a country

#### gigi9

) IF the rate of increase o f population of a country is 3 percent every year, by what factor does it increase every 10 years? What percentage increase will double the population every ten years?
***N= Pe^(k*t) <---this is the equation, not sure how to use it....

#### HallsofIvy

Are you sure this doesn't belong in "homework"?

Given N= Pekt, then since e0= 1, P must be the population for the initial year. You are told that the population increases by 3% each year, so after one year it must be
1.03P: N(t)= 1.03P= Pek(1). You can divide by P to get 1.03= ek and so k= ln(1.03).

You can now write the formula as N(t)= P e(ln(1.03))t
but it would be a really good idea to note that ekt= (ek)t and since eln(1.03)= 1.03 the equation is just N(t)= P*(1.03)t.

After 10 years, you have either N(10)= Pe(10ln(1.03)) or
N(10)= P*(1.03)10. The "factor" by which it increases is
that number multiplying P: e(10ln(1.03))= (1.03)10.

Notice that that "1.03" is precisely because the population was increasing by 3%= 0.03 each year. If the percentage increase was 100r%, then the factor would be 1+r. To answer the second part, "What percentage increase will double the population every ten years?", solve either Pe10ln(1+r)= 2P or P(1+r)10= 2P for r.

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