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= Pe^kt, 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= Pe^k(1). You can divide by P to get 1.03= e^k 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 e^kt= (e^k)^t and since e^ln(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 Pe^10ln(1+r)= 2P or P(1+r)¹⁰= 2P for r.

 

[M98Ranger]

The rate of increase of population of a country is a crucial factor in determining the growth and development of a nation. A 3 percent annual increase in population may seem small, but it can have significant implications in the long run.

To answer the first question, we can use the equation N= Pe^(k*t), where N is the final population, P is the initial population, k is the growth rate, and t is the time in years. In this case, we can assume that the initial population (P) is 1, and the growth rate (k) is 3 percent (0.03). So, the equation becomes N= e^(0.03*t).

To find the factor by which the population increases every 10 years, we can substitute t= 10 in the equation and solve for N. This gives us N= e^(0.03*10) = e^0.3 ≈ 1.349. Therefore, the population of the country will increase by a factor of 1.349 every 10 years.

To determine the percentage increase that will double the population every 10 years, we can use the concept of exponential growth. In this case, we want the population to double every 10 years, which means N= 2P. Substituting this in the equation, we get 2P= e^(0.03*t). Solving for t, we get t= ln(2)/0.03 ≈ 23.1 years. Therefore, a 3 percent increase in population every year will double the population every 23.1 years, which is slightly more than double the required 10 years.

In conclusion, a 3 percent annual increase in population may not seem significant, but it can have a considerable impact on the population growth of a country. It is essential for governments to monitor and manage the population growth rate to ensure sustainable development and well-being of their citizens.