Chapter 3, Problem 27CT

To determine

Find the exponential growth model that fits the points shown in the graph.

Answer to Problem 27CT

The exponential model for the given graph is $y=2745e^{0.1570x}$

Explanation of Solution

Given:

The graph:

  

Let’s assume the exponential model for the given graph is $y=a{e}^{hx}$ . The given graph passes through the points (0,2745) and $(9,11277).$ So, the function $y=a{e}^{hx}$ should satisfy the points. put the point (0,2745) in the function,

  $2745=a{e}^{bw}$

  $a=2745$

Putting the value of the variable $a$ in the function,

  $y=2745e^{bx}$

Now, put the point (9,11277) in the function,

  $11277=2745e^{b\times 9}$

  $e^{b_{\infty }9}=\frac{11277}{2745}$

Taking the natural logarithms on both sides,

  $\ln e^{\ln \alpha }=\ln \frac{11277}{2745}$

Using the property of natural logarithm,

  $b\times 9=\ln \frac{11277}{2745}$

  $b=\frac{1}{9}\ln \frac{11277}{2745}$

  $b=0.1570$

Putting the value of the variable $b$ in the function,

  $y=2745e^{0.1570x}$

"Hence, the exponential model for the given graph is $y=2745e^{0.1570x}$