Chapter 3.2, Problem 4E

Solution

To decide: whether the function is an exponential growth function or exponential

decay function, and find the constant percentage rate of growth or decay.

Exponential decay function

Constant percentage rate of decay = 0.32%

Given information:

The function:

  $f\left(x\right)=5607\cdot {0.9968}^x$

Definition Used:

Exponential Population Model

If a population P is changing at a constant percentage rate r each year, then

  $P\left(t\right)=P_0{\left(1+r\right)}^t$   ..... (1)

Where $P_0$ is the initial population, r is expressed as a decimal, and t is time in years.

If $r>0$, then P ( t ) is an exponential growth function, and its growth factor is the base of the exponential function, $1+r$ .

On the other hand, if $r<0$, the base $1+r<1$, P ( t ) is an exponential decay function, and $1+r$ is the

decay factor for the population.

Explanation:

The given function can be rewritten as:

  $\begin{array}{c}f\left(x\right)=5607\cdot {0.9968}^x\\ f\left(x\right)=5607{\left(1+\left(-0.0032\right)\right)}^t\end{array}$

Comparing it with the general form (1), it is observed that here,

  $r=-0.0032$

Which is less than 0, so, f is an exponential decay function with a decay rate of 0.32%.