Chapter 9.7, Problem 26P

The population of a city for years since 2000 is shown below. Which kind of function best models the data? Write an equation to model the data.

To determine

To find : Which type of function best model the given data, use differences or ratio.

Answer to Problem 26P

Exponential model is the most appropriate model and the equation to model the data is $y=1500{\left(2\right)}^x$ .

Explanation of Solution

Given information :

The population of a city for years since 2000 is shown in table $\begin{array}{cccccc}\text{Years since 2000}& 0& 2& 4& 6& 8\\ \text{Population}& 1500& 6000& 24,000& 96,000& 384,000\end{array}$

Calculation :

To find the type of model from the given set of data use the followings:

1. If the first difference of the y values is common, then linear model fits the data
2. If the second difference of the y values is common, then quadratic model fit the data.
3. If the y values have the common ratio, then an exponential model fits he data.

For the given set of data, find the find difference of the y values:

  $\begin{array}{l}6000-1500=4500\\ 24000-6000=18000\\ 96000-24000=72000\\ 384000-96000=288000\end{array}$

Since, the first differences is not common, so find the second differences

  $\begin{array}{l}18000-4500=13500\\ 72000-18000=54000\\ 288000-72000=216000\end{array}$

Since, the second differences are also not common, so find the ratio of the y values.

  $\begin{array}{l}\frac{6000}{1500}=4\\ \frac{24000}{6000}=4\\ \frac{96000}{24000}=4\\ \frac{384000}{96000}=4\end{array}$

Since, y values have the common ratio. Therefore, exponential model is the most appropriate model.

Let the exponential model be

  $y=a\cdot b^x$ ...... (1)

Substitute $\left(0,1500\right)$ and find a:

  $\begin{array}{l}1500=a\cdot b^0\\ 1500=a\cdot 1\\ a=1500\end{array}$

Substitute $\left(2,6000\right)$ and find b:

  $\begin{array}{l}6000=1500\cdot b^2\\ \frac{6000}{1500}=b^2\\ 4=b^2\\ 2=b\end{array}$

Therefore, the equation is $y=1500{\left(2\right)}^x$

Hence,

Exponential model is the most appropriate model and the equation to model the data is $y=1500{\left(2\right)}^x$ .