Chapter 6.7, Problem 25E

To determine

To show : An exponential model should be the good fit for the original data pairs $\left(x,y\right)$ and then write an exponential model for the original data.

Answer to Problem 25E

The exponential function that models the data is $y=\left(4.13\right){\left(1.05\right)}^x$ .

Explanation of Solution

Given information :

The given data is $\begin{array}{cccccc}x\left(\text{in years}\right)& 20& 30& 40& 50& 60\\ y\left(\text{in centimeters}\right)& 12& 15& 25& 40& 100\end{array}$ .

Calculation :

Create a table of data pairs $\left(x,\text{ ln}y\right)$ .

  $\begin{array}{cccccc}x& 20& 30& 40& 50& 60\\ \ln y& 2.48& 2.70& 3.23& 3.67& 4.60\end{array}$

Plot the above transformed points as shown below:

  

Since, the point lie close to a line, so an exponential model should b a good fit for the original data.

Find an exponential model $y=a{b}^x$ by choosing any two points on the line, such as $\left(20,2.48\right)$ and $\left(60,\text{ 4}\text{.60}\right)$ . Use these points to write an equation of the line. Then solve for y.

Therefore,

  $\begin{array}{l}\ln y-2.48=0.053\left(x-20\right)\text{ (Equation of the line)}\\ \ln y=0.053x+1.42\text{ (Simplify)}\\ y=e^{0.053x+1.42}\text{ (Exponentiate each side using base e)}\\ y=e^{0.053x}\left(e^{1.42}\right)\text{ (Use peoperty of exponents)}\\ y=\left(4.13\right){\left(1.05\right)}^x\text{ (Simplify)}\end{array}$

Hence,

The exponential function that models the data is $y=\left(4.13\right){\left(1.05\right)}^x$ .