Chapter 7.5, Problem 37E

a.

To determine

To find: a curve to model the data and superimpose it on a scatter plot of population against years after 1950 using logistic regression.

Answer to Problem 37E

Explanation of Solution

Given information: Population Growth Table shows the population of Laredo, Taxes for selected years between 1950 and 2003.

Years after 1950	Population
0	10,571
20	81,437
30	138,857
40	180,650
50	215,794
53	218,027

Calculation:

Using logistic regression utility, a curve the model of the above table data is:

  $P=\frac{233620}{1+14e^{-0.1t}}$ .

The graph of the above model using graphing utility is shown below.

  

b.

To determine

To find: what number will the Laredo population approach in the long run based on the regression equation.

Answer to Problem 37E

233620.

Explanation of Solution

Given information: $P=\frac{233620}{1+14e^{-0.1t}}$ .

Calculation:

  $\underset{t\to \infty }{\lim }P=\underset{t\to \infty }{\lim }\frac{233620}{1+14e^{-0.1t}}=233620$ .

c.

To determine

To find: when would the Laredo population first exceed 225,000 people based on the regression equation.

Answer to Problem 37E

In 2006, the Laredo population first exceeds 225,000 people based on the regression equation.

Explanation of Solution

Given information: $P=\frac{233620}{1+14e^{-0.1t}}$

Calculation:

  $\begin{array}{c}P=\frac{233620}{1+14e^{-0.1t}}\\ P=225000\\ 225000=\frac{233620}{1+14e^{-0.1t}}\Rightarrow t=59\end{array}$

Therefore, in 1990 +59 = 2006, the Laredo population first exceeds 225,000 people based on the regression equation.

d.

To determine

To write: a logistic differential equation in the form $dP/dt=kP\left(M-P\right)$ that models the growth of the Laredo data in given Table.

Answer to Problem 37E

  $\frac{dP}{dt}=0.1P(233620-P)$ .

Explanation of Solution

Given information: $P=\frac{233620}{1+14e^{-0.1t}}$

Calculation:

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

  $\frac{dP}{dt}=0.1P(233620-P).$