Chapter 8.8, Problem 3CYU

(a)

To determine

To Find: The graph of the function $P\left(t\right)=\frac{16500}{1+18e^{-0.085t}}$ for $0\le t\le 200$ .

Explanation of Solution

Consider the given function is,

  $P\left(t\right)=\frac{16500}{1+18e^{-0.085t}}$

The time range is $0\le t\le 200$ .

The graph for the given function is shown in Figure 1

  

Figure 1

(b)

To determine

To Find: The horizontal asymptote.

Answer to Problem 3CYU

The vertical asymptote is $P\left(t\right)=16500$ .

Explanation of Solution

From the graph shown in Figure 1 the value y of the graph is approaching 16500.

Thus, the vertical asymptote is $P\left(t\right)=16500$ .

(c)

To determine

To Find: The maximum population.

Answer to Problem 3CYU

The maximum value of population is 16500.

Explanation of Solution

From part (b) and from the figure 1 the since the graph approaches 16500 with the value of y the maximum value of population is 16500.

(d)

To determine

To Find: The time when the population reaches 16450.

Answer to Problem 3CYU

The number of years to reach the population 16450 is 102 years.

Explanation of Solution

Consider the given function is,

  $P\left(t\right)=\frac{16500}{1+18e^{-0.085t}}$

Then,

  $\begin{array}{l}16450=\frac{16500}{1+18e^{-0.085t}}\\ 1+18e^{-0.085t}=1.003\\ e^{-0.085t}=\frac{0.003}{18}\\ \ln e^{-0.085t}=\ln \left(1.666\times {10}^{-4}\right)\end{array}$

Then,

  $\begin{array}{l}-0.085t=-8.69\\ t=\frac{-8.69}{-0.085}\\ t=102.235\end{array}$

The number of years to reach the population 16450 is 102 years.