Respuesta :
Answer:
a) The population in 2020 will be of 7.14 million people.
b) It will take 9.23 years for the population to drop below 7 million people
Step-by-step explanation:
Exponential equation for population decay:
The equation that models a population in exponential decay, after t years, is given by:
[tex]P(t) = P(0)(1-r)^t[/tex]
In which P(0) is the initial population and r is the decay rate, as a decimal.
The total population of a European country is decreasing at a rate of 0.6% per year.
This means that [tex]r = 0.006[/tex]
In 2014, the population of the country was 7.4 million people.
This means that [tex]P(0) = 7.4[/tex]. Then
[tex]P(t) = P(0)(1-r)^t[/tex]
[tex]P(t) = 7.4(1-0.006)^t[/tex]
[tex]P(t) = 7.4(0.994)^t[/tex]
a) What is the population likely to be in 2020 if it decreases at the same rate?
2020 - 2014 = 6, so this is P(6).
[tex]P(6) = 7.4(0.994)^6 = 7.14[/tex]
The population in 2020 will be of 7.14 million people.
b) How long will it take for the population to drop below 7 million people?
t when P(t) = 7. So
[tex]P(t) = 7.4(0.994)^t[/tex]
[tex]7 = 7.4(0.994)^t[/tex]
[tex](0.994)^t = \frac{7}{7.4}[/tex]
[tex]\log{(0.994)^t} = \log{\frac{7}{7.4}}[/tex]
[tex]t\log{0.994} = \log{\frac{7}{7.4}}[/tex]
[tex]t = \frac{\log{\frac{7}{7.4}}}{\log{0.994}}[/tex]
[tex]t = 9.23[/tex]
It will take 9.23 years for the population to drop below 7 million people
