Question

The number of people living in a country is increasing each year exponentially so that the number of people 5 years ago was 15,000. The number of people now is 26,000. What is the present population of the country? What equation can we use to describe the situation, if we let the growth rate be represented by r? Solve the equation for r. What is the growth rate? What will the population be 5 years from today?

Answer 1

The present population of the country is 26,000.

`y=a(1+r)^(x)`

The growth rate is approximately 12%.

The population 5 years from today will be 45,821.

Explanation:

(1)

The present population of the country is 26,000.

(2)

It is provided that the number of people living in a country is increasing each year exponentially.

Then the exponential growth function can be used to describe the situation.

The exponential growth function is:

`y=a(1+r)^(x)`

Here,

y = final value

a = initial value

r = growth rate

x = time

(3)

It is provided that x = 5 years ago the population was a = 15,000 and at present the population is y = 26,000.

Compute the value of r as follows:

`y=a(1+r)^(x)`

`26000=15000* (1+r)^(5)`

`(1+r)=[(26000)/(15000)]^(1/5)`

`1+r=1.1163`

`r=0.1163\\r\approx 0.12`

Thus, the growth rate is approximately 12%.

(4)

Compute the population 5 years from today as follows:

`y=a(1+r)^(x)`

`=26000* (1+0.12)^(5)\\\\=26000* 1.762342\\\\=45820.892\\\\\approx 45821`

Thus, the population 5 years from today will be 45,821.