The population of a country was 8.4 million people in 2000. Three years later, the population of the same nation was 9.3 million people. Assuming that population is growing exponentially, what is the projected population (rounded to one-tenth of a million) of the country in 2008?
Just plug the numbers into the good old eponential growth formula:
P(t) = Po * e^rt
P(0) = 8.4 so Po = 8.4
P(t) = 8.4 e^rt
P(3) = 9.3 = 8.4 e^3r
9.3/8.4 = e^3r
ln 1.107 = 3r
r = ln 1.107/3 = 0.0339
P(t) = 8.4 e^.0339t
P(8) = 8.4 e^0.271 = 11.0
To solve this problem, we can use the exponential growth formula:
P(t) = P0 * e^(rt)
Where:
P(t) is the population at time t,
P0 is the initial population,
r is the growth rate, and
t is the time in years.
Given that the population in 2000 (t=0) was 8.4 million (P0), the population in 2003 (t=3) was 9.3 million (P(3)).
Let's plug those values into the formula to find the growth rate (r):
9.3 = 8.4 * e^(3r)
Divide both sides of the equation by 8.4:
e^(3r) = 9.3 / 8.4
Simplify the right side:
e^(3r) = 1.10714285714
Take the natural logarithm (ln) of both sides to solve for r:
3r = ln(1.10714285714)
Divide both sides of the equation by 3:
r = ln(1.10714285714) / 3
Using a calculator, calculate ln(1.10714285714) and divide the result by 3:
r ≈ 0.03472163
Now that we have the growth rate (r), let's find the projected population in 2008 (t=8).
P(8) = P0 * e^(8r)
Plugging in the values:
P(8) = 8.4 * e^(8 * 0.03472163)
Calculate 8 * 0.03472163:
P(8) = 8.4 * e^(0.27777304)
Using a calculator, calculate e^(0.27777304), and multiply the result by 8.4:
P(8) ≈ 8.4 * 1.31917099
P(8) ≈ 11.06179106
Therefore, the projected population of the country in 2008 is approximately 11.1 million people.
To estimate the projected population of the country in 2008, we can use the exponential growth formula:
P(t) = P0 * e^(rt)
Where:
P(t) is the projected population at time t
P0 is the initial population
e is the mathematical constant approximately equal to 2.71828
r is the growth rate
t is the time elapsed in years
First, we need to find the growth rate (r). We can use the formula:
r = ln(P1/P0) / (t1 - t0)
Where:
P1 is the final population
P0 is the initial population
t0 is the initial time
t1 is the final time
For this problem, P1 = 9.3 million, P0 = 8.4 million, t0 = 2000, and t1 = 2003.
Calculating the growth rate:
r = ln(9.3/8.4) / (2003 - 2000)
r = ln(1.1071) / 3
r ≈ 0.0361
Now, we can use the growth rate to estimate the population in 2008 (t = 8).
P(t) = P0 * e^(rt)
P(8) = 8.4 * e^(0.0361*8)
P(8) ≈ 8.4 * e^(0.2888)
P(8) ≈ 8.4 * 1.3346
P(8) ≈ 11.23 million
Therefore, the projected population of the country in 2008 (rounded to one-tenth of a million) is approximately 11.2 million people.