Chapter 7.5, Problem 38E

a.

To determine

To find a curve to model the data and superimpose it on a scatter plot of population against tears after 1950.

Answer to Problem 38E

The curve is,

  

Explanation of Solution

Given information: Data

  

Calculation: The curve is,

  

  $P=\frac{459310}{1+18e^{-0.11t}}$

b.

To determine

To find what number will the Virginia Beach population approach in the long run?

Answer to Problem 38E

The Virginia Beach population is 459310.

Explanation of Solution

Given information: Data

  

Calculation: The Virginia Beach population is obtained as,

  $\begin{array}{l}P=\frac{459310}{1+18e^{-0.11t}}\\ =459310\end{array}$

Hence, the Virginia Beach population is 459310.

c.

To determine

To find when would the Virginia Beach population first exceed 450,000 people?

Answer to Problem 38E

In 2010 the Virginia Beach population first exceed 450,000 people.

Explanation of Solution

Given information: Data

  

Calculation: The required solution is obtained as,

  $\begin{array}{l}450000=\frac{459310}{1+18e^{-0.11t}}\\ t=60\end{array}$

that is 2010.

Hence, In 2010 the Virginia Beach population first exceed 450,000 people.

d.

To determine

To write a logistic differential equation in the form $dP/dt=kP\left(M-P\right)$ .

Answer to Problem 38E

The required solution is $\frac{dP}{dt}=0.11P\left(459310-P\right)$ .

Explanation of Solution

Given information:

  

Calculation: The required solution is obtained as,

Using the long-term carrying capacity from part (b) and the growth rate from the logistic regression in part (a), we plug those into the logistic differential equation to find:

  $\frac{dP}{dt}=0.11P\left(459310-P\right)$

Hence, the required solution is $\frac{dP}{dt}=0.11P\left(459310-P\right)$ .

e.

To determine

To write a logistic differential equation in the form $dP/dt=kP\left(M-P\right)$ .

Answer to Problem 38E

The required solution is $\frac{dP}{dt}=0.11P\left(459310-P\right)$ .

Explanation of Solution

Given information:

  

Calculation: The required solution is obtained as,

Using the long-term carrying capacity from part (b) and the growth rate from the logistic regression in part (a), we plug those into the logistic differential equation to find:

  $\frac{dP}{dt}=0.11P\left(459310-P\right)$

Hence, the required solution is $\frac{dP}{dt}=0.11P\left(459310-P\right)$ .