Question

The world's population was 5.51 billion on January 1, 1993 and 5.88 billion on January 1, 1998. Assume that at any time the population grows at a rate proportional to the population at that time. In what year should the world's population reach 8 billion

Answer 1

Answer:

The world's population should reach 8 billion during the year 2021.

Step-by-step explanation:

The world population can be modeled by the following equation:

$\frac{dP}{dt} = r$

It's solution is:

$P(t) = P(0)e^{rt}$

In which P(t) is the population in t years ater 1993, in billions of people, P(0) is the initial population(in 1993) and r is the growth rate.

5.51 billion on January 1, 1993

This means that P(0) = 5.51.

5.88 billion on January 1, 1998.

1998 - 1993 = 5

This means that P(5) = 5.88

We use this as a mean to find the value of r.

$P(t) = P(0)e^{rt}$

$5.88 = 5.51e^{5r}$

$e^{5r} = \frac{5.88}{5.51}$

$e^{5r} = 1.06715$

$\ln{e^{5r}} = \ln{1.06715}$

$5r = \ln{1.06715}$

$r = \frac{\ln{1.06715}}{5}$

$r = 0.013$

Assume that at any time the population grows at a rate proportional to the population at that time. In what year should the world's population reach 8 billion

t yers after 1993, in which t is found when P(t) = 8. So

$P(t) = 5.51e^{0.013t}$

$8 = 5.51e^{0.013t}$

$e^{0.013t} = \frac{8}{5.51}$

$e^{0.013t} = 1.4519$

$\ln{e^{0.013t}} = \ln{1.4519}$

$0.013t = \ln{1.4519}$

$t = \frac{\ln{1.4519}}{0.013}$

$t = 28.68$

1993 + 28.68 = 2021.68

The world's population should reach 8 billion during the year 2021.