Chapter 7.5, Problem 20PPS

a.

To determine

To estimate the number of rabbits in March.

Answer to Problem 20PPS

207

Explanation of Solution

Given:

Population of rabbit increases according to the model,

  $p=200{(2.5)}^{0.02t}$ where $t$ is the number of months and $t=0$ corresponds to January.

Calculation:

Given that the model shows the population of rabbit

  $p=200{(2.5)}^{0.02t}$

If $t=0$ corresponds to January then $t=2$ corresponds to march.

Therefore, the population of rabbit in march is,

  $\begin{array}{l}$ \Rightarrow p=200{(2.5)}^{0.02(2)}$$ \Rightarrow p=200{(2.5)}^{0.04}$\Rightarrow p=200(1.037)\\ \Rightarrow p=207.4\simeq 207\end{array}$

Conclusion:

In the month of March the population of the rabbits is 207.

b.

To determine

To graph the function $p=200{(2.5)}^{0.02t}$ , find the domain, range and equation of the asymptote.

Explanation of Solution

Given:

Function: $p=200{(2.5)}^{0.02t}$

Calculation for graph:

Consider $p=200{(2.5)}^{0.02t}$

Values of x	Values of p
0	500
1	230.6
2	207.46
-1	196.36

By taking different values of x , the graph can be plotted.

Graph:

  

Interpretation:

Domain:

The domain of the expression $p=200{(2.5)}^{0.02t}$ is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Hence,

Domain: $(-\infty ,\infty )$

Range is the set of all y -values that the function passes through.

Hence,

Range: $(0,\infty )$ for all y, $y>0$

Asymptotes:

Exponential functions have only horizontal asymptotes.

So, there will be no vertical asymptotes, since the function do not have a denominator.

Therefore, equation of horizontal asymptote is $y=0.$