Chapter 6, Problem 7CA

To determine

Exponential model that represent the given data sets.

Answer to Problem 7CA

Exponential model for data set (a) is $y_a=5\times 2^x$

Exponential model for data set (b) is $y_b=\frac{1}{2}\times 3^x$

Explanation of Solution

Given information:

  

These data sets represents exponential model.

Formula used:

$x$01234$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:

  

From the graph (a) we get

$x$$2$$3$$4$$y$$20$$40$$80$

So, common ratio= $r=\frac{40}{20}=2$

So, we can write $y=a_0{2}^x$

Now, $x=2;y=20$

  $\begin{array}{l}\Rightarrow 20=a_0{2}^x\\ \Rightarrow a_0=\frac{20}{4}=5\end{array}$

So, $y=5\times 2^x$

  

Common ratio= $r=\frac{13.5}{4.5}=3$

Therefore, $y=a_0{3}^x$

Now, $x=2;y=4.5$

  $\begin{array}{l}4.5=a_0{3}^2\\ \Rightarrow a_0=\frac{4.5}{9}=\frac{1}{2}\end{array}$

So, $y=\frac{1}{2}\times 3^x$

We assume the $y$ for data (a) as $y_a$ and for data set (b) as $y_b$

Therefore, $y_a=5\times 2^x$ and $y_b=\frac{1}{2}\times 3^x$