Problem 44 - Calculus Early Transcendentals: Pearson New International Edition

Question

GG 44. The Census Bureau estimates that the growth rate \(k\) of the world population will decrease by roughly \(0.0002\) per year for the next few decades. In \(2004, k\) was \(0.0132\). (a) Express \(k\) as a function of time \(t\), where \(t\) is measured in years since 2004 . (b) Find a differential equation that models the population \(y\) for this problem. (c) Solve the differential equation with the additional condition that the population in \(2004(t=0)\) was \(6.4\) billion. (d) Graph the population \(y\) for the next 300 years. (e) With this model, when will the population reach a maximum? When will the population drop below the 2004 level?

Step-by-Step Solution

Answer

(a) \( k(t) = 0.0132 - 0.0002t \); (b) \( \frac{dy}{dt} = (0.0132-0.0002t)y \); (c) \( y(t) = 6.4e^{0.0132t - 0.0001t^2} \); (d) Graph showing rise and decline; (e) Max in 2070, below 2004 level in 2136.

Step 1: Define k as a function of time

Given that the growth rate \( k \) decreases by \( 0.0002 \) per year starting from \( k = 0.0132 \) in 2004, we can express \( k \) as a function of time \( t \). It can be written as: \[ k(t) = 0.0132 - 0.0002t \] where \( t \) is measured in years since 2004.

Step 2: Find the differential equation for population y

The rate of change of population \( y \) can be modeled by the differential equation \( \frac{dy}{dt} = k(t)y \). Substituting \( k(t) \) derived from Step 1, we get: \[ \frac{dy}{dt} = (0.0132 - 0.0002t)y \].

Step 3: Solve the differential equation

Solve the equation \( \frac{dy}{dt} = (0.0132 - 0.0002t)y \) using the method of separation of variables. First, separate variables: \[ \frac{dy}{y} = (0.0132 - 0.0002t) dt \]Integrate both sides: \[ \ln y = 0.0132t - 0.0001t^2 + C \]Exponentiate to solve for \( y \): \[ y(t) = e^{0.0132t - 0.0001t^2 + C} \]Using the initial condition \( y(0) = 6.4 \), we find \( C \): \[ 6.4 = e^C \implies C = \ln 6.4 \]Thus, the solution is: \[ y(t) = 6.4e^{0.0132t - 0.0001t^2} \].

Step 4: Graph the population function y(t)

Plot the function \( y(t) = 6.4e^{0.0132t - 0.0001t^2} \) over the interval from \( t = 0 \) to \( t = 300 \). The graph will show the initial growth followed by a decline as \( t \) increases.

Step 5: Determine when the population reaches a maximum

Find the maximum point by setting the derivative of \( y(t) \) with respect to \( t \) equal to zero and solving for \( t \):\[ y'(t) = y(t)(0.0132 - 0.0002t) \] The critical point occurs when \( 0.0132 - 0.0002t = 0 \): \[ t = \frac{0.0132}{0.0002} = 66 \].Thus, the population reaches a maximum in the year \( 2004 + 66 = 2070 \).

Step 6: Determine when population falls below 2004 level

Solve for \( t \) when \( y(t) < 6.4 \) by setting the equation: \[ 6.4e^{0.0132t - 0.0001t^2} < 6.4 \] Simplifying gives: \[ e^{0.0132t - 0.0001t^2} < 1 \] Taking the natural logarithm of both sides gives: \[ 0.0132t - 0.0001t^2 < 0 \] Factoring, we find that the population falls below the 2004 level when \( t > 132 \), i.e., the year 2004 + 132 = 2136.

Key Concepts

Population Growth Model

Understanding the population growth model is crucial for analyzing how populations change over time. In this context, the model is established based on the Census Bureau's estimation that the population growth rate will gradually decrease. At the heart of this model are several key components:

• **Growth Rate (\( k \))**: Initially set at 0.0132 in 2004, this rate decreases by 0.0002 per year. This linear decrease forms the core of the model, as it predicts changes in population dynamics over time.
• **Function of Time (\( t \))**: Time is measured from the year 2004, making it easier to understand and predict future trends using the model.

With this structure, the population growth rate can be expressed as a function of time: \[ k(t) = 0.0132 - 0.0002t \]. This allows us to predict changes in population size and analyze potential future trends. By recognizing this decreasing growth rate, we can understand that the population initially increases before starting to slow down due to the decrement in the growth rate.

Separation of Variables

Separation of variables is a mathematical technique often used to solve differential equations. When we apply this method to the problem of population growth, it becomes a powerful tool for finding explicit solutions for how the population will evolve over time. Here's how it works in this context:

• **Step 1: Formulate the Equation**
  The differential equation for population growth \( \frac{dy}{dt} = k(t)y \), where \( k(t) = 0.0132 - 0.0002t \).
• **Step 2: Separate Variables**
  Move all terms involving \( y \) to one side, and those involving \( t \) to the other: \[ \frac{dy}{y} = (0.0132 - 0.0002t) dt \].
• **Step 3: Integrate Both Sides**
  This step results in \[ \ln y = 0.0132t - 0.0001t^2 + C \], where \( C \) is the constant of integration.
• **Step 4: Solve for \( y \) **
  Finally, exponentiate to solve for \( y \): \[ y = e^{0.0132t - 0.0001t^2 + C} \]. Utilizing initial conditions can help find the value of constant \( C \).

Separation of variables simplifies solving differential equations by partitioning the variables, thus making integration possible and finding explicit solutions more straightforward.

Exponential Functions

Exponential functions form the backbone of many growth and decay models, including population growth. Their key characteristic is that the rate of change of a quantity is proportional to the quantity itself. In our population model, the solution takes the form of an exponential function:

• **Form**
  The general solution derived from our problem is \( y(t) = 6.4e^{0.0132t - 0.0001t^2} \). This exemplifies how populations can rapidly increase at first, then stabilize or even decrease as time progresses.
• **Significance**
  Exponential growth captures real-world scenarios where populations grow proportionally to their size, reflecting resource availability and environmental constraints in adding realism to predictions.
• **Behavior**
  These functions can reflect various growth scenarios from constant exponential growth to inhibition due to decaying rates or resources, providing flexible modeling of phenomena.

The understanding of exponential functions is crucial as they depict a realistic picture of how populations might behave if certain parameters, like growth rates, change over time, offering insights into long-term sustainable development trends.

Census Bureau Data

Utilizing Census Bureau data provides a foundational perspective on population studies and models. Accurate data serves as a benchmark for developing predictive models and understanding demographic trends. Here are some of the reasons why this data is invaluable:

• **Reliable Source**
  Being a respected national entity, the Census Bureau provides reliable and detailed data which can be used for modeling. This data ensures that predictions and analyses are grounded in reality.
• **Forecasting Trends**
  With insights such as growth rate predictions, policymakers and researchers benefit from anticipating potential demographic changes, aiding in resource planning and policy-making.
• **Historical Insights**
  By analyzing historical growth data, researchers can compare past trends with current 'decrease in growth rate' predictions to understand shifts in demographic patterns over time.

Census Bureau data guides us in establishing realistic and informed population models, allowing for thoughtful planning, projections, and discussion around future demographic challenges and opportunities.