Chapter 6, Problem 15CT

To determine

To find: The exponential model for the given data set and then to find the year when the snow mobile in worth $2500.

Answer to Problem 15CT

  $y=4200\times {\left(0.9\right)}^x$

After $4.924$ years worth the snow mobile is $ $2500$ .

Explanation of Solution

Given information:

  

Formula used:

$x$	0	1	2	3	4
$y$	$a_0$	$a_1$	$a_2$	$a_3$	$a_4$

If the consecutive ratios of the y-values are all equal to a common value $r$ , then $y$ can be modeled by an exponential function.

  $r=\frac{a_{n+1}}{a_n}\text{ exponential ratio}$

  $y=a_0{r}^n\text{ exponential model}$

Calculation:

  $x=0\text{ ; }y=4200$

So, $a_0=4200$

  $r=\frac{3780}{4200}=0.9$

So, $y=4200\times {\left(0.9\right)}^x$

Now, we have to find for which value of $x$ , $y=2500$

  $\begin{array}{l}2500=4200{\left(0.9\right)}^x\\ \Rightarrow \frac{2500}{4200}={\left(0.9\right)}^x\\ \Rightarrow \log \left(\frac{25}{42}\right)=x\log \left(0.9\right)\\ \Rightarrow x=\frac{\log \left(\frac{25}{42}\right)}{\log \left(0.9\right)}\\ \Rightarrow x=4.924\end{array}$

So, after $4.924$ years worth the snow mobile is $ $2500$ .