A country's population in 1995 was 175 million. In 2000 it was 181 million. Estimate the population in 2020 using the exponential growth formula. Round you answer to the nearest million.
P=Ae^kt
207
To estimate the population in 2020 using the exponential growth formula P=Ae^kt, we need to first find the values of A, k, and t.
Given that the population in 1995 was 175 million (P = 175) and in 2000 it was 181 million (P = 181), we can set up two equations:
1) 175 = A * e^(k * 0) (population in 1995)
2) 181 = A * e^(k * 5) (population in 2000)
From the first equation, we can see that A * e^(k * 0) = 175, which simplifies to A = 175.
Substituting this value of A in the second equation, we get:
181 = 175 * e^(k * 5)
Dividing both sides by 175:
e^(k * 5) = 181/175
To find the value of k, we will take the natural logarithm (ln) of both sides of the equation, since e^k is the base of natural logarithm:
ln(e^(k * 5)) = ln(181/175)
Using the logarithmic property of ln(e^x) = x:
k * 5 = ln(181/175)
Dividing both sides by 5:
k = ln(181/175) / 5
k ≈ 0.00732
Now that we have the values for A and k, we can use them to estimate the population in 2020 (t = 25 years from 1995 to 2020).
P = Ae^(kt)
P = 175 * e^(0.00732 * 25)
P ≈ 236
Rounding the estimate to the nearest million, the population in 2020 would be approximately 236 million.
175e^5k = 181
so, k = .00674
P = 175 e^(.00674t)
Now plug in t=25.