Chapter 3.5, Problem 46E

a.

To determine

To use : a graphing utility to graph the model

Explanation of Solution

Given information : The question is

The number $Y$ of yeast organism in a culture is given by the model $Y=\frac{663}{1+72e^{-00547t}}$

Graph : Sketch the graph using graphing utility.

Step 1: Press WINDOW button to access the Window editor.

Step 2: Press $\boxed{\text{Y=}}$ button.

Step 3: Enter the expression $\frac{663}{1+72e^{-00547t}}$ which is required to graph.

Step 4: Press $\boxed{\text{GRAPH}}$ button to graph the function.

The graph is obtained as:

  

Interpretation: from the above graph it is observed that the function $Y=\frac{663}{1+72e^{-00547t}}$ is $0\le t\le 18$

b.

To determine

To predict : the population of $19th$ hour and $30th$ hour

Answer to Problem 46E

The population of $19th$ hour is $661.54$ and $30th$ hour is $663$

Explanation of Solution

Given information : The question is

The number $Y$ of yeast organism in a culture is given by the model $Y=\frac{663}{1+72e^{-00547t}}$

Calculation :

  $\begin{array}{l}Y=\frac{663}{1+72e^{-00547t}}\\ p\left(19\right)=\frac{663}{1+72e^{-00547\left(19\right)}}\Rightarrow 661.54\\ p\left(30\right)=\frac{663}{1+72e^{-00547\left(30\right)}}\Rightarrow 663\end{array}$

c.

To determine

What is the limiting value of the population

Answer to Problem 46E

The limiting value of the population is

Explanation of Solution

Given information : The question is

The number $Y$ of yeast organism in a culture is given by the model $Y=\frac{663}{1+72e^{-00547t}}$

Calculation :

The horizontal asymptote are $p\left(t\right)=663$ is the limiting value of function as $t$ is increasing $p\left(t\right)$ remains constant

d.

To determine

Why do you think this population of yeast follows a logistic growth model instead of an exponential growth model

Answer to Problem 46E

The curve follows the logistic growth model

Explanation of Solution

Given information : The question is

The number $Y$ of yeast organism in a culture is given by the model $Y=\frac{663}{1+72e^{-00547t}}$

Calculation :

The population of yeast in a culture is determined by its capacity and as the population rises the death also rise and hence the curve follows the logistic growth model