In 1990, the population of Africa was 643 million and by 2000 it had grown to 813 million.
a.) Use the exponential growth model A=A0 e^kt, in which t is the number of years after 1990, to find the exponential growth function that models the data.
b.) By which year will Africa population reach 2 billion?
643 e^10k = 813
e^10k = 813/643 = 1.2644
10k = ln 1.2644 = 0.2346
k = 0.02346
now just solve for t in
643 e^0.02346t = 2x10^9
a.) To find the exponential growth function that models the data, we need to find the values of A0 and k in the equation A = A0 e^(kt).
Given information:
In 1990, the population of Africa was 643 million (A0).
By 2000, the population had grown to 813 million (A).
Let's assign 0 to the year 1990 (t = 0) and 10 to the year 2000 (t = 10) since there are 10 years between 1990 and 2000.
Using the given data, we can set up two equations:
Equation 1: 643 million = A0 e^(0k) -- since e^0 is 1, this simplifies to A = A0
Equation 2: 813 million = A0 e^(10k)
To solve for A0 and k, we can divide Equation 2 by Equation 1:
(813 million) / (643 million) = (A0 e^(10k)) / A0
1.264 = e^(10k)
Taking the natural logarithm of both sides, we get:
ln(1.264) = ln(e^(10k))
ln(1.264) = 10k * ln(e)
ln(1.264) = 10k * 1
ln(1.264) = 10k
Next, we can solve for k by dividing both sides by 10:
k = ln(1.264) / 10
Using a calculator, we find:
k ≈ 0.02598
Therefore, the exponential growth model that represents the data is:
A = A0 e^(0.02598t)
b.) To find the year when Africa's population reaches 2 billion, we can set up the exponential growth model:
A = A0 e^(0.02598t)
Given information:
A = 2 billion (2,000 million)
A0 = 643 million
k ≈ 0.02598
Setting A equal to 2,000 million, we get:
2,000 million = 643 million e^(0.02598t)
Divide both sides by 643 million:
3.108 = e^(0.02598t)
Taking the natural logarithm of both sides:
ln(3.108) = ln(e^(0.02598t))
ln(3.108) = 0.02598t * ln(e)
ln(3.108) = 0.02598t * 1
ln(3.108) = 0.02598t
Next, we can solve for t by dividing both sides by 0.02598:
t = ln(3.108) / 0.02598
Using a calculator, we find:
t ≈ 31.599
Therefore, Africa's population is projected to reach 2 billion in approximately 31.6 years after 1990, which would be around the year 2021.