Question

The total population of a European country is decreasing at a rate of 0.6% per year. In 2014, the population of the country was 7.4 million people.
a) What is the population likely to be in 2020 if it decreases at the same rate?

b) How long will it take for the population to drop below 7 million people?

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Answer 1

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:

$P(t) = P(0)(1-r)^t$

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 $r = 0.006$

In 2014, the population of the country was 7.4 million people.

This means that $P(0) = 7.4$. Then

$P(t) = P(0)(1-r)^t$

$P(t) = 7.4(1-0.006)^t$

$P(t) = 7.4(0.994)^t$

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).

$P(6) = 7.4(0.994)^6 = 7.14$

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

$P(t) = 7.4(0.994)^t$

$7 = 7.4(0.994)^t$

$(0.994)^t = \frac{7}{7.4}$

$\log{(0.994)^t} = \log{\frac{7}{7.4}}$

$t\log{0.994} = \log{\frac{7}{7.4}}$

$t = \frac{\log{\frac{7}{7.4}}}{\log{0.994}}$

$t = 9.23$

It will take 9.23 years for the population to drop below 7 million people