The population of the world was about $5.3$ billion in 1990 . Birth rates in the 1990 s range from 35 to 40 million per year and death rates range from 15 to 20 million per year. Let's assume that the carrying capacity for world population is 100 billion. Use the logistic model to predict the world population in the 2,450 year. Calculate your answer in billions to one decimal place. (Because the initial population is small compared to the carrying capacity, you can take $k$ to be an estimate of the initial relative growth rate.)
A) $78.3$ billion
B) $27.0$ billion
C) $17.1$ billion
D) $59.2$ billion
E) $32.9$ billion

Question

Quizplus · Accepted Answer

The logistic growth model is given by $ P(t) = \frac{K}{1 + \frac{K-P_0}{P_0}e^{-kt}} $, where $ K $ is the carrying capacity, $ P_0 $ is the initial population, and $ k $ is the growth rate. The initial growth rate $ k $ can be estimated as the net growth rate, which is the birth rate minus the death rate divided by the initial population: $ k = \frac{(35-15) \text{ million}}{5.3 \text{ billion}} \approx 0.0038 $. Using this in the logistic model with $ K = 100 $ billion, $ P_0 = 5.3 $ billion, and $ t = 2450 - 1990 = 460 $ years, we find $ P(460) \approx 27.0 $ billion.

The Population of the World Was About 5.35.35.3 Billion in 1990 kkk

The Population of the World Was About $5.3$ Billion in 1990 $k$