A country's population in 1991 was 15 million. In 2001 it was 16 million. Estimate the population in 2005 using exponential growth formula. Round your answer to the nearest million.
let N = 15 e^(kt), where N is in million, t is in years since 1991 and k is a constant
when t = 10 , N = 16
16 = 15e^(10k)
1.0666667 = e^(10k)
ln 1.0666667 = 10k
k = .006456852
then when t = 14
N = 15e^(14(.00645852)) = 16.418
So to the nearest million it would still be 16 million
To estimate the population in 2005 using the exponential growth formula, we can assume that the population growth rate remains constant. The formula for exponential growth is:
P(t) = P0 * (1 + r)^t
Where:
P(t) is the population at time t
P0 is the initial population
r is the growth rate
t is the time in years
Given that the population in 1991 (t=0) was 15 million (P0), and in 2001 (t=10) it was 16 million (P(10)), we can find the growth rate (r) and use it to estimate the population in 2005 (t=14).
To find the growth rate, we can use the formula:
r = (P(10) / P0)^(1/t) - 1
Substituting the given values:
r = (16 million / 15 million)^(1/10) - 1
Calculating the growth rate:
r = 1.06666666667^(0.1) - 1
r = 0.06495137956
Now, substituting the values into the exponential growth formula for t=14:
P(t) = P0 * (1 + r)^t
P(14) = 15 million * (1 + 0.06495137956)^14
P(14) ≈ 15 million * (1.06495137956)^14
P(14) ≈ 15 million * 1.145569
Calculating the population in 2005:
P(14) ≈ 17.184 million
Rounding the population to the nearest million:
P(14) ≈ 17 million
Therefore, the estimated population in 2005 using the exponential growth formula is approximately 17 million.
To estimate the population in 2005 using the exponential growth formula, we need to know the growth rate and the initial population.
The formula for exponential growth is: P(t) = P(0) * e^(rt), where:
- P(t) is the population at time t
- P(0) is the initial population
- e is the base of the natural logarithm (approximately 2.71828)
- r is the growth rate
- t is the time in years
First, let's calculate the growth rate (r):
r = ln(P(1)/P(0))/10, where ln is the natural logarithm function.
P(1) is the population in 2001, which is 16 million.
P(0) is the population in 1991, which is 15 million.
r = ln(16/15)/10
r ≈ 0.0460517
Now, we can use this growth rate to estimate the population in 2005 (t = 14 years from 1991 to 2005):
P(2005) ≈ P(0) * e^(rt)
P(0) is the population in 1991, which is 15 million.
t = 14
r ≈ 0.0460517
P(2005) ≈ 15 * e^(0.0460517 * 14)
Calculating this expression will give us the estimated population in 2005, rounding the answer to the nearest million.