Question

a country's population and the growth rate in 2011 are given below. assume the growth rate remains the same from year to year. Use this information and an exponential model to estimate the country's population in the specified yearpopulation = 105 milliongrowth rate = -0.657%year, 2024the size of the population in 2024 will be about ____ million (round to the nearest hundredth as needed)

Answer 1

A general exponential model can be described by:

`P(x)=A\cdot b^x`

Where A is the initial condition, and b is the growth factor. From the problem, P is the population (in millions); if x is the number of years passed since 2011, then x = 0 in 2011:

`105=A\cdot b^0\Rightarrow A=105`

We know that the growth rate is -0.657% = -0.00657. The growth rate r and the growth factor b are related by:

`b=1+r\Rightarrow b=1+0.00657=0.99343`

Our model is now complete:

`P(x)=105\cdot0.99343^x`

For the year 2024, x = 13 years have passed. Then, we calculate the population using our model (rounding to the nearest hundredth):

`P(x=13)=105\cdot0.99343^(13)=96.38`

The size of the population in 2024 will be about 96.38 million