The population of a particular country was 29 million in 1985; in 1997, it was 38 million. The exponential growth function describes the population of this country t years after 1985. Use the fact that 12 years after 1985 the population increased by 9 million to find k to three decimal places
p(t) = ce^(kt)
p(0) = 29, so c=29
p(12) = 38
29e^(12k) = 38
Now just solve for k
To find the value of k in the exponential growth function, we can use the given information that the population increased by 9 million after 12 years (in 1997).
Let's denote the initial population in 1985 as P(0) and the population 12 years later in 1997 as P(12).
We are given that P(0) = 29 million, P(12) = 38 million, and we want to find the value of k.
The general formula for exponential growth is P(t) = P(0) * e^(kt), where P(t) represents the population at time t, P(0) is the initial population, e is Euler's number (approximately 2.71828), and k is the rate of growth.
We can rewrite the exponential growth formula in terms of our given values:
P(12) = P(0) * e^(k * 12)
Substituting the given values:
38 million = 29 million * e^(k * 12)
Now, let's isolate the exponential term:
e^(k * 12) = 38 million / 29 million
Simplifying:
e^(k * 12) ≈ 1.31034
To find the value of k, we need to take the natural logarithm (ln) of both sides of the equation:
ln(e^(k * 12)) ≈ ln(1.31034)
Using the property ln(e^x) = x:
k * 12 ≈ ln(1.31034)
Dividing both sides by 12:
k ≈ ln(1.31034) / 12
Using a calculator, we find:
k ≈ 0.023
Therefore, the value of k to three decimal places is approximately 0.023.