Chapter 6.7, Problem 19E

To determine

To write : A function that models the data.

Answer to Problem 19E

The exponential function that models the data is $y=\left(\frac{81}{14}\right){\left(\frac{14}{9}\right)}^x$ .

Explanation of Solution

Given information :

A store sells motorized scooters. The table shows the number of y of scooters sold during the xth year that the store has been open.

Calculation :

Make a scatter plot of the data. The scatter plot of the data is drawn below:

  

Therefore, the data appears exponential.

Choose any two points to write a model, such as $\left(1,9\right)$ and $\left(2,\text{ 14}\right)$ . Substitute the coordinate of these points into $y=a{b}^x$ . Therefore,

  $9=a{b}^1$ …… (1)

  $14=a{b}^2$ ….. (2)

Divide equation (2) by equation (1) and solve:

  $\begin{array}{l}\frac{14}{9}=\frac{a{b}^2}{a{b}^1}\\ \frac{14}{9}=b\end{array}$

Substitute $b=\frac{14}{9}$ in equation (1) and solve for a:

  $\begin{array}{l}9=\frac{14}{9}a\\ \frac{81}{14}=a\end{array}$

So, the exponential function that models the data is $y=\left(\frac{81}{14}\right){\left(\frac{14}{9}\right)}^x$

Hence,

The exponential function that models the data is $y=\left(\frac{81}{14}\right){\left(\frac{14}{9}\right)}^x$ .