Chapter 3.2, Problem 54E

Solution

To decide: the constant percentage rate of decay of $P\left(t\right)=22.7\cdot {0.834}^t$ .

B

Given information:

The options:

1. 22.7%
2. 16.6%
3. 8.34%
4. 2.27%
5. 0.834%

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:

  $\begin{array}{c}P\left(t\right)=22.7\cdot {0.834}^t\\ P\left(t\right)=22.7{\left(1+\left(-0.166\right)\right)}^t\end{array}$

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

  $r=-0.166$

Which is less than 0, so, P is an exponential decay function with a decay rate of 16.6%.

Correct choice is Option B.