Estimating Population Growth with Logistic Model

Math Problem Statement

Homework:HW SECTION 9.5 Question 8, 9.5.38 Part 5 of 7 HW Score: 94.64%, 7.57 of 8 points Points: 0.57 of 1

Skip to Main content Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 Question 7 Question 8 Question content area top Part 1 A nation's population (to the nearest million) was 281 million in 2000 and 311 in 2010. It is projected that the population in 2050 will be 437 million. To construct a logistic model, both the growth rate and the carrying capacity must be estimated. There are several ways to estimate these parameters. Use parts (a) through (f) to use one approach. Question content area bottom Part 1 a. Assume that tequals0 corresponds to 2000 and that the population growth is exponential for the first ten years; that is, between 2000 and 2010, the population is given by Upper P left parenthesis t right parenthesis equals Upper P left parenthesis 0 right parenthesis e Superscript rt. Estimate the growth rate r using this assumption. requals

0.01014 (Round to five decimal places as needed.) Part 2 b. Write the solution of the logistic equation with the value of r found in part (a). Write any populations in the logistic equation in millions of people. P(t)equals

StartFraction negative Upper K times 281 Over e Superscript negative 0.01014 t Baseline left parenthesis negative Upper K plus 281 right parenthesis minus 281 EndFraction Part 3 Use the projected value Upper P left parenthesis 50 right parenthesis equals 437 million to find a value of the carrying capacity K. Kequals

2744.39 (Type an integer or decimal rounded to the nearest hundredth as needed.) Part 4 c. According to the logistic model determined in parts (a) and (b), when will the country's population reach 95% carrying capacity? The population will reach 95% of the carrying capacity in the year

2504.47. (Type an integer or decimal rounded to the nearest hundredth as needed.) Part 5 d. Estimations of this kind must be made and interpreted carefully. Suppose the projected population for 2050 is 450 million rather than 437 million. What is the value of the carrying capacity in this case? Kequals

enter your response here (Type an integer or decimal rounded to the nearest hundredth as needed.) r(Round to five decimal places as needed.)P(t)K(Type an integer or decimal rounded to the nearest hundredth as needed.)The population will reach 95% of the carrying capacity in the year(Type an integer or decimal rounded to the nearest hundredth as needed.)K(Type an integer or decimal rounded to the nearest hundredth as needed.) Get more help

Solution

To solve part (d), we need to find the new value of the carrying capacity $K$ when the projected population in 2050 is 450 million instead of 437 million.

Step 1: Use the Logistic Model Formula

The general form of the logistic equation is:

$P(t) = \frac{K \cdot P_0}{P_0 + (K - P_0) e^{-rt}}$

where:

$P(t)$ is the population at time $t$ ,
$P_0$ is the initial population,
$r$ is the growth rate,
$K$ is the carrying capacity.

From the previous parts, we know:

$P(50) = 450$ million (new population projection for 2050),
$t = 50$ years (since $t = 0$ corresponds to the year 2000),
$P_0 = 281$ million (population in 2000),
$r = 0.01014$ (growth rate obtained in part a).

Step 2: Plug in the Known Values

Substitute the known values into the logistic equation to solve for $K$ :

$450 = \frac{K \times 281}{281 + (K - 281) e^{-0.01014 \times 50}}$

Step 3: Solve for $K$

Simplify and solve for $K$ :

$450 = \frac{K \times 281}{281 + (K - 281) e^{-0.507}}$

Calculate $e^{-0.507}$ :

$e^{-0.507} \approx 0.602$

So, the equation becomes:

$450 = \frac{K \times 281}{281 + 0.602(K - 281)}$

Now, solve for $K$ :

$450(281 + 0.602(K - 281)) = 281K$

Expanding and solving the equation:

$450 \times 281 + 450 \times 0.602K - 450 \times 0.602 \times 281 = 281K$

Simplify:

$126450 + 270.9K - 75850.68 = 281K$

$50599.32 = 10.1K$

$K = \frac{50599.32}{10.1} \approx 5019.73$

Final Answer

The new carrying capacity $K$ when the projected population for 2050 is 450 million is approximately 5019.73 million.

Do you have any questions or would you like more details on the steps? Here are five questions related to this problem:

How does changing the growth rate $r$ affect the carrying capacity $K$ ?
What would be the carrying capacity if the growth rate $r$ were different?
How does the logistic model differ from an exponential growth model?
Why does the logistic model assume a carrying capacity?
How would the model change if population projections are given for different years?

Tip: The logistic model is useful for predicting populations that approach a limit over time, unlike exponential models which predict unlimited growth.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Logistic Growth Model
Exponential Growth
Population Dynamics

Formulas

Logistic Equation: P(t) = K * P0 / (P0 + (K - P0) * e^(-rt))
Exponential Growth: P(t) = P0 * e^(rt)

Theorems

Logistic Growth Theorem

Suitable Grade Level

Grades 11-12

Related Recommendation

Logistic Growth Model: Estimating Population Growth and Carrying Capacity

Predicting Population Growth Using a Logistic Model with Earth's Carrying Capacity

Estimating US Population Growth Using Logistic Model

Logistic Population Growth Model: Estimating Earth's Population in 2054

Predicting Population Growth with Logistic Equation