Problem 70 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

CURBING POPULATION GROWTH Five years ago, the government of a Pacific Island state launched an extensive propaganda campaign toward curbing the country's population growth. According to the Census Department, the population (measured in thousands of people) for the following 4 yr was $$ P(t)=-\frac{1}{3} t^{3}+64 t+3000 $$ where \(t\) is measured in years and \(t=0\) corresponds to the start of the campaign. Find the rate of change of the population at the end of years \(1,2,3\), and 4 . Was the plan working?

Step-by-Step Solution

Answer

The rate of change of the population at the end of years 1, 2, 3, and 4 are 63, 60, 55, and 48, respectively. Since the rate of change is decreasing as time passes, the plan to curb population growth is indeed working.

Step 1: Find the derivative of P(t) with respect to t.

To find the rate of change of the population with respect to time, we need to find the derivative of P(t) with respect to t. This will give us a function that describes how the population is changing as time passes. To find the derivative of P(t), use the power rule for differentiation: \(P(t) = -\frac{1}{3}t^3 + 64t + 3000\) \(\frac{dP}{dt} = -t^2 + 64\)

Step 2: Evaluate the derivative at t = 1, 2, 3, and 4.

Now, we need to evaluate the derivative of P(t) at t = 1, 2, 3, and 4 to find the rate of change of the population at each of these times. \(\frac{dP}{dt}(1) = -(1)^2 + 64 = 63\) \(\frac{dP}{dt}(2) = -(2)^2 + 64 = 60\) \(\frac{dP}{dt}(3) = -(3)^2 + 64 = 55\) \(\frac{dP}{dt}(4) = -(4)^2 + 64 = 48\)

Step 3: Determine if the plan is working.

To determine if the plan to curb population growth is working, we need to see if the rate of change of the population is decreasing as time goes on. From the evaluated derivatives, we can see that: At t = 1, the rate of change is 63. At t = 2, the rate of change is 60. At t = 3, the rate of change is 55. At t = 4, the rate of change is 48. Since the rate of change of the population is decreasing as time passes, we can conclude that the plan to curb population growth is working.

Key Concepts

Understanding Rate of Change

Discussing the rate of change in population models is akin to understanding the speed at which populations are growing or shrinking over time. It essentially answers the question: how quickly is the number of individuals in a population increasing or decreasing?

The rate of change is found by taking the first derivative of the population function with respect to time. In the given exercise, the population function, represented as P(t), is differentiated to give us a new function. This new function describes the velocity of the population's growth or decline; it tells us by how many thousands of people the population changes each year.

Evaluating the Significance of the Rate of Change

• If the derivative is positive, the population is growing.
• If the derivative is negative, the population is shrinking.
• If the derivative decreases over time, even if it remains positive, it indicates that the growth rate of the population is slowing down.

Thus, to assess the impact of the government's efforts in the exercise, we needed to compute this derivative and evaluate it at specific points in time.

Deriving the Population Model

In the exercise, the derivative of the population model reveals how the country's population policy is influencing the number of people. Calculus, and particularly differentiation, provides a powerful tool for modeling such complex scenarios.

Differentiation involves applying rules like the power rule: for a term ax^n, the derivative is anx^(n-1). Applying this to the given population model P(t), we obtained the derivative P'(t), which simplifies to -t^2 + 64. This derivative function then allowed us to evaluate the rate of change for successive years.

Interpreting the Derivative

• A positive value suggests population growth.
• A decreasing positive value indicates a slowing growth rate.
• A thorough examination of these values over time gives insights into the effectiveness of government policies.

Calculus in Demography

Applying calculus in demography is pivotal for interpreting and predicting demographic changes over time. Demographers use calculus to understand the dynamics of population data, providing insights for policy making and social planning.

In the provided problem, calculus enabled the examination of a population policy's effectiveness over a defined period. By calculating the rate of change year-by-year, we established a trend line of the population's growth rate. The successive decrease in the derivative's values indicated a decelerating population growth, thus signaling the potential effectiveness of the government's campaign.

The Broader Implications of Demographic Calculus

• It aids in forecasting future population sizes.
• Mathematical models can inform public health, urban development, and resource management.
• Calculus-driven models help in understanding the social impact of various demographic trends, such as aging population or migration.