Chapter 3.2, Problem 53E

Solution

To decide: the constant percentage rate of growth of $P\left(t\right)=1.23\cdot {1.049}^t$ .

C

Given information:

The options:

1. 49%
2. 23%
3. 4.9%
4. 2.3%
5. 1.23%

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:

  $\begin{array}{c}P\left(t\right)=1.23\cdot {1.049}^t\\ P\left(t\right)=1.23{\left(1+0.049\right)}^t\end{array}$

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

  $r=0.049$

Which is greater than 0, so, P is an exponential growth function with a growth rate of 0.049 or 4.9%.

Correct choice is Option C.