Chapter 6, Problem 38CR

To determine

To write : exponential model for the data pairs $\left(x,y\right)$

Answer to Problem 38CR

The exponential function is $\ln y=\left(1.34\right){\left(1.22\right)}^x$

Explanation of Solution

Given information :

The exponential model is

$x$	1	2	3	4
$\ln y$	$1.64$	$2.00$	$2.36$	$2.72$

Calculation :

The solution of the exponential model is

Step 1 The 1 given table from four values of $x$ and $\ln y$ . Convent into co-ordinates

  $\left(1,1.64\right),\left(2,2.00\right),\left(3,2.36\right),\left(4,2.72\right)$ .

Step 2 Substitute the co- ordinates of the four points into

  $\begin{array}{l}\ln y=a{b}^x\to \text{equation}\left(1\right)\\ 1.64=a{b}^{{}^1}\to \text{equation}\left(2\right)\\ 2=a{b}^2\to \text{equation}\left(3\right)\\ 2.36=a{b}^3\to \text{equation}\left(4\right)\\ 2.72=a{b}^4\to \text{equation}\left(5\right)\end{array}$

Step 3 Solve for a is $\text{equation}\left(2\right)$ to obtain $a=\frac{1.64}{b}$ and substitute this expression for a in $\text{equation}\left(3\right)$

  $\begin{array}{l}2=\frac{1.64}{b}{b}^2\\ 2=1.64b^{2-1}\\ 2=1.64b\\ b=\frac{2}{1.64}\\ b=1.22\end{array}$

Step 4 Determine that

  $\begin{array}{l}a=\frac{1.64}{b}\\ a=\frac{1.64}{1.22}\\ a=1.34\end{array}$

Hence, the exponential function is $\ln y=\left(1.34\right){\left(1.22\right)}^x$