# The u S population in 1910 was 92 million people. In 1990 the population was 250 million. How do you use the information to create both a linear and an exponential model of the population?

Feb 14, 2017

#### Explanation:

The linear model means that there is a uniform increase and in this case of US population from $92$ million people in $1910$ to $250$ million people in $1990$.

This means an increase of $250 - 92 = 158$ million in $1990 - 1910 = 80$ years or

$\frac{158}{80} = 1.975$ million per year and in $x$ years it will become

$92 + 1.975 x$ million people. This can be graphed using the linear function $1.975 \left(x - 1910\right) + 92$,
graph{1.975(x-1910)+92 [1890, 2000, 85, 260]}

The exponential model means that there is a uniform proportional increase i.e. say p% every year and in this case of US population from $92$ million people in $1910$ to $250$ million people in $1990$.

This means an increase of $250 - 92 = 158$ million in $1990 - 1910 = 80$ years or

p% given by $92 {\left(1 + p\right)}^{80} = 250$ which gives us ${\left(1 + p\right)}^{80} = \frac{250}{92}$ which simplifies to $p = {\left(\frac{250}{92}\right)}^{0.0125} - 1 = 0.0125743$ or 1.25743%.

This can be graphed as an exponential function $92 \times {1.0125743}^{\left(x - 1910\right)}$, which gives population in a year $y$ and this appears as
graph{92(1.0125743^(x-1910)) [1900, 2000, 85, 260]}