Question

At the beginning of a population study, the population of a large city was 1.65 million people. Three years later, the population was 1.74 million people. Assume that population grows according to an uninhibited exponential growth model.

What would be the population of the city 5 years after the start of the population study?

A. 2.1518
B. 1.7683
C. 1.8027
D. 1.8843
F. None of the Above

Answer 1

C. 1.8027

Explanation:

The exponential population growth model is given by:

`P(t) = P_(0)e^(rt)`

In which P(t) is the population after t years,
`P_(0)` is the initial population and r is the growth rate.

At the beginning of a population study, the population of a large city was 1.65 million people. Three years later, the population was 1.74 million people.

This means that
`P_(0) = 1.65, P(3) = 1.74`

Applying this to the equation, we find r. So

`P(t) = P_(0)e^(rt)`

`1.74 = 1.65e^(3r)`

`e^(3r) = (1.74)/(1.65)`

`e^(3r) = 1.0545`

Applying ln to both sides

`\ln{e^(3r)} = ln(1.0545)`

`3r = ln(1.0545)`

`r = (ln(1.0545))/(3)`

`r = 0.0177`

So

`P(t) = 1.65e^(0.0177t)`

What would be the population of the city 5 years after the start of the population study?

This is P(5).

`P(t) = 1.65e^(0.0177t)`

`P(5) = 1.65e^(0.0177*5)`

`P(5) = 1.8027`

So the correct answer is:

C. 1.8027