Chapter 6.3, Problem 7Q

To determine

To determine whether the function represents exponential growth or exponential decay

Also identify the percent rate of change

Answer to Problem 7Q

The function represents an exponential growth function.

Percent rate of change = 26%

Explanation of Solution

Given information:

Given function

  $f\left(t\right)=\frac{1}{3}{\left(1.26\right)}^t$

Concept used:

A function of the form $y=a{\left(1+r\right)}^t,wherea>0\text{ and }r>0and\left(1+r\right)>0$ , is an exponential growth function.

A function of the form $y=a{\left(1-r\right)}^t,wherea>0\text{ and 0< }r<1,\left(1-r\right)<1$ , is an exponential decay function.

( where y is final amount, initial amount, r is rate of decay (decay factor) and t is time )

Comparing given function $f\left(t\right)=\frac{1}{3}{\left(1.26\right)}^t$ with standard function

  $1.26>1$

So, this is case of exponential growth

  $f\left(t\right)=\frac{1}{3}{\left(1.26\right)}^t$ is an exponential growth function

Hence,

  $\begin{array}{l}\left(1+r\right)=1.26\\ r=1.26-1=0.26\end{array}$

Percent of decay $=0.26\times 100=26\%$