Chapter 4.2, Problem 27E

World Population The relative growth rate of world population has been decreasing steadily in recent years. On the basis of this, some population models predict that world population will eventually stabilize at a level that the planet can support. One such logistic model is

$P(t)=\frac{73.2}{6.1+5.9e^{-0.02t}}$

where t = 0 is the year 2000 and population is measured in billions.

1. (a) What world population does this model predict for the year 2200? For 2300?
2. (b) Sketch a graph of the function P for the years 2000 to 2500.
3. (c) According to this model, what size does the world population seem to approach as time goes on?

To determine

To find: The world population predict for year 2000 and 2003.

Answer to Problem 27E

The world population in the year 2200 is $11.79\text{billions}$ and in the year 2300 is $11.97\text{billions}$.

Explanation of Solution

Given:

The world population follows the logistic growth model

$P\left(t\right)=\frac{73.2}{6.1+5.9e^{-0.02t}}$

Calculations:

The world population follows the logistic growth model

$P\left(t\right)=\frac{73.2}{6.1+5.9e^{-0.02t}}$

Substitute 200 for t in the above formula to calculate prediction of world population in year 2200,

$\begin{array}{c}P\left(200\right)=\frac{73.2}{6.1+5.9e^{-0.02\times 200}}\left[\because e=2.718\right]\\ =\frac{73.2}{6.1+5.9\times {2.718}^{-4}}\\ =\frac{73.2}{6.20}\\ =11.79\text{billions}\end{array}$

Hence, the world population in the year 2200 is $11.79\text{billions}$.

Substitute 300 for t in the above formula to calculate prediction of world population in year 2300,

$\begin{array}{c}P\left(300\right)=\frac{73.2}{6.1+5.9e^{-0.02\times 300}}\left[\because e=2.718\right]\\ =\frac{73.2}{6.1+5.9\times {2.718}^{-6}}\\ =\frac{73.2}{6.11}\\ =11.97\text{billions}\end{array}$

Hence, the world population in the year 2300 is $11.97\text{billions}$.

Hence, the world population in the year 2200 is $11.79\text{billions}$ and in the year 2300 is $11.97\text{billions}$.

To determine

To plot: The graph of the function $P\left(t\right)$ for the year 2000 to 2500.

Explanation of Solution

The world population follows the logistic growth model

$P\left(t\right)=\frac{73.2}{6.1+5.9e^{-0.02t}}$

The population at $t=2000,2100,2200,2300,2400,2500$ is shown in table (1),

t	$P\left(t\right)$ in billons
0	6.11
100	10.61
200	11.79
300	11.97
400	11.99
500	11.99

Connect all the resulting point with smooth curve,

The graph of the function $P\left(t\right)$ for the year 2000 to 2500 is shown Figure (1),

Figure (1)

To determine

To evaluate: The world population bird if time goes on.

Answer to Problem 27E

The world population if time goes on is $12\text{billons}$.

Explanation of Solution

Given:

The world population follows the logistic growth model

$P\left(t\right)=\frac{73.2}{6.1+5.9e^{-0.02t}}$

Calculation:

The world population follows the logistic growth model

$P\left(t\right)=\frac{73.2}{6.1+5.9e^{-0.02t}}$

Substitute $\infty$ for t in the above formula to calculate the world population if time goes on,

$\begin{array}{c}P\left(\infty \right)=\frac{73.2}{6.1+5.9e^{-0.02\times \infty }}\left[\because e=2.718\right]\\ =\frac{73.2}{6.1+5.9\times {2.718}^{-\infty }}\\ =\frac{73.2}{6.1}\\ =12\text{billons}\end{array}$

Hence, the world population if time goes on is $12\text{billons}$.