(a) Use the fact that the population was 250 million in 1990 (t = 0) to formulat
ID: 2893930 • Letter: #
Question
(a) Use the fact that the population was 250 million in 1990 (t = 0) to formulate a logistic model for the population of a particular region. (Assume the carrying capacity is 4200 million.)
(b) Determine the value of k in your model by using the fact that the population in 2000 was 275 million. (Round your answer to eight decimal places.)
(c) Use your model to predict the population in the years 2100 and 2200. (Round your answers to the nearest whole number.)
(d) Use your model to predict the year in which the population will exceed 350 million.
Explanation / Answer
Use P(t) = P(0)e^(kt) where t is the number of years since 1990.
Population in 1990 is given as 250 million, so P(0) = 250
Population in 2000 is given as 275 million, so P(10) = 275
275 = 250e^(10k)
e^(10k) = 275/250 = 1.1
10k = ln(1.1)
10k = 0.0953101798
k = 0.00953102
So P(t) = 250e^(0.00953102t)
Population in year 2100....P(110) = 713.279335 million
Population in year 2200....P(210) = 1.85 billion
To find the year that the population exceeds 350 million......
250e^(0.00953102t) 350
e^(0.00953102t) 350 / 250
0.00953102t ln(1.4)
t ln(1.4) / 0.00953102
t ln(1.4) / 0.00953102
t 35.30285705
According to this model, the population should exceed 350 million approximately 35.3 years after 1990, or sometime in the year 2025
Related Questions
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.