Assume the exponential growth model a(t)equalsupper a 0 e superscript kt and a world population of 6.3 billion in 2004. if the population must stay below 25 billion during the next 100 years, what is the maximum acceptable annual rate of growth?

Hello there!

We knot that the model is of the form A(t) = A*e^(kt)

We also know that in 2004 the population was 6.3 billion and that in the next 100 years the population needs to be less than 25 billion. We want to find the maximum acceptable annual rate of growth.

Let's define t = 0 at 2004, this means that

A(0) = 6.3 billions = A*e^(k*0) = A

then A = 6.3 billions

and for finding the maximum k, we need to do the next step:

A(100) = 25 billion = (6.3 billion)*e^(k*100)

now we need to solve it for k:

25 billion = (6.3 billion)*e^(k*100)

25 = 6.3*e^(k*100)

e^(k*100) = 25/6.3  

now we can aply the natural logaritm in both sides:

k*100 = ln(25/6.3)

k = ln(25/6.3)/100 = 0.014

So we know that if he population must stay below 25 billion during the next 100 years, then we need to use k < 0.014