Problem 47 - Calculus Early Transcendentals: Pearson New International Edition

Question

Using the same axes, draw the graphs for \(0 \leq t \leq 100\) of the following two models for the growth of world population (both described in this section). (a) Exponential growth: \(y=6.4 e^{0.0132 t}\) (b) Logistic growth: \(y=102.4 /\left(6+10 e^{-0.030 t}\right)\) Compare what the two models predict for world population in 2010,2040, and 2090 . Note: Both models assume that world population was \(6.4\) billion in \(2004(t=0)\).

Step-by-Step Solution

Answer

The exponential model predicts faster population growth compared to the logistic model which stabilizes.

Step 1: Understand the given models

We are provided with two mathematical models to describe world population growth: 1. Exponential Growth Model: \( y = 6.4 e^{0.0132t} \) 2. Logistic Growth Model: \( y = \frac{102.4}{6 + 10e^{-0.030t}} \)Both models are functions of time \( t \) where \( t = 0 \) represents the year 2004, and \( y \) is the population in billions. For visualization, we need to draw both curves on the same set of axes for \( t \) ranging from 0 to 100.

Step 2: Set up the graphing equations

Prepare to graph both models for \( 0 \leq t \leq 100 \). This corresponds to a time range from 2004 to 2104. We'll evaluate and plot the exponential and logistic growth equations over this interval.- Exponential curve: \( y = 6.4 e^{0.0132t} \)- Logistic curve: \( y = \frac{102.4}{6 + 10e^{-0.030t}} \)

Step 3: Calculate values for specific years

Calculate the population predicted by both models for the years 2010 \((t=6)\), 2040 \((t=36)\), and 2090 \((t=86)\).### For Exponential growth:- 2010: \( y = 6.4 e^{0.0132 \times 6} \)- 2040: \( y = 6.4 e^{0.0132 \times 36} \)- 2090: \( y = 6.4 e^{0.0132 \times 86} \)### For Logistic growth:- 2010: \( y = \frac{102.4}{6 + 10e^{-0.030 \times 6}} \)- 2040: \( y = \frac{102.4}{6 + 10e^{-0.030 \times 36}} \)- 2090: \( y = \frac{102.4}{6 + 10e^{-0.030 \times 86}} \)

Step 4: Perform calculations

### Exponential Model- 2010: \( y = 6.4 e^{0.0792} \approx 6.4 \times 1.0823 = 6.923 \text{ billion} \)- 2040: \( y = 6.4 e^{0.4752} \approx 6.4 \times 1.6089 = 10.299 \text{ billion} \)- 2090: \( y = 6.4 e^{1.1352} \approx 6.4 \times 3.1128 = 19.971 \text{ billion} \)### Logistic Model- 2010: \( y = \frac{102.4}{6 + 10e^{-0.180}} \approx \frac{102.4}{6 + 8.3471} = 8.563 \text{ billion} \)- 2040: \( y = \frac{102.4}{6 + 10e^{-1.080}} \approx \frac{102.4}{6 + 3.437} = 11.777 \text{ billion} \)- 2090: \( y = \frac{102.4}{6 + 10e^{-2.580}} \approx \frac{102.4}{6 + 0.757} = 12.583 \text{ billion} \)

Step 5: Compare model predictions

Compare the calculated values for the years 2010, 2040, and 2090 between the two growth models: - **2010**: Exponential Growth ≈ 6.923 billion; Logistic Growth ≈ 8.563 billion. - **2040**: Exponential Growth ≈ 10.299 billion; Logistic Growth ≈ 11.777 billion. - **2090**: Exponential Growth ≈ 19.971 billion; Logistic Growth ≈ 12.583 billion. The exponential model shows a faster increase in population over time compared to the logistic model, which slows down, approaching a carrying capacity.

Key Concepts

Exponential Growth

Exponential growth is a mathematical concept in which a quantity increases by a constant percentage over equal time intervals. For population, it means that as the population grows, it adds more people rapidly because the growth is proportional to the size of the existing population. This model is described using the formula \( y = y_0 e^{rt} \), where \( y_0 \) is the initial population, \( r \) is the growth rate, and \( t \) is time.
For example, using the exponential growth model from the exercise, \( y = 6.4 e^{0.0132t} \), represents world population growth. The term \( 6.4 \) is the initial population in billions in the year 2004 (\( t=0 \)), and \( 0.0132 \) is the growth rate.

• This model predicts significant population increases as time goes on, such as from 6.923 billion in 2010 to 19.971 billion in 2090.
• It assumes no limiting factors such as resource constraints, thus potentially leading to unrealistic predictions over long periods.

The exponential model is often utilized in theoretical scenarios where boundless growth is assumed, although real-world constraints can alter its applicability.

Logistic Growth

Logistic growth provides a more realistic representation of population growth by incorporating limiting factors, such as available resources. Instead of growing indefinitely, logistic growth assumes that a population will grow rapidly at first, then slow as it approaches a carrying capacity, which is the maximum population size that can be sustainably supported. The logistic growth formula is given by \( y = \frac{K}{1 + A e^{-rt}} \), where \( K \) is the carrying capacity, \( A \) is a constant related to the initial population, \( r \) is the growth rate, and \( t \) is time.
In the exercise, the logistic growth model is \( y = \frac{102.4}{6 + 10 e^{-0.030t}} \). Here, the carrying capacity is 102.4 billion, reflecting a theoretical cap on the world population.

• In 2010, the model predicted a population of 8.563 billion, moving to 12.583 billion by 2090.
• The logistic model tends to level off over time, suggesting a stabilization of the population as it nears the carrying capacity.

Compared to exponential growth, logistic growth provides a more nuanced view that considers environmental and resource constraints, making it a valuable tool for long-term population predictions.

World Population Prediction

World population prediction involves estimating future populations based on different assumptions and models. The exercise explores two such models providing contrasting outcomes: exponential and logistic growth.
Assumptions made in these models significantly affect predictions.

• Exponential growth forecasts boundless increase whereas logistic growth forecasts a stabilization due to resource limitations.
• Predictions for 2010, 2040, and 2090 showed exponential growth leading to much larger population numbers compared to the more conservative logistic model.

Population predictions guide global planning and policy-making by providing insights into potential challenges related to resources, infrastructure, and environmental impact. Developing accurate models is thus crucial for anticipating future societal needs and guiding sustainable development.

Graphical Analysis

Graphical analysis involves plotting mathematical models to visually compare predictions and understand trends. In population studies, such graphs highlight the significant differences in predictions by various models.
By plotting exponential and logistic growth models on the same axis, we can observe:

• The steep upward trajectory of exponential growth, depicting exponential increases over time.
• The S-shaped curve of logistic growth showing rapid initial growth that slows and plateaus as it nears a carrying capacity.

Comparing these models graphically for given timeframes (e.g., 2010, 2040, 2090) helps in visualizing the effects of growth rates and environmental constraints on population predictions. This approach aids in understanding long-term implications for global population dynamics and helps in assessing model suitability for policy planning and resource management.