Problem 370 - Calculus Volume 1

Question

For the next set of exercises, use the following table, which features the world population by decade. $$ \begin{array}{|l|l|} \hline \text { Years since 1950 } & \text { Population (millions) } \\ \hline 0 & 2,556 \\ \hline 10 & 3,039 \\ \hline 20 & 3,706 \\ \hline 30 & 4,453 \\ \hline 40 & 5,279 \\ \hline 50 & 6,083 \\ \hline 60 & 6,849 \\ \hline \end{array} $$ [T] The best-fit exponential curve to the data of the form \(P(t)=a e^{b t}\) is given by \(P(t)=2686 e^{0.01604 t}\). Use a graphing calculator to graph the data and the exponential curve together.

Step-by-Step Solution

Answer

Plot each data point and the curve in a graphing calculator to see their alignment.

Step 1: Understand the Data

The table provided shows the world population in millions for years since 1950. You need to interpret these points as coordinates where 'Years since 1950' represents the x-axis and 'Population (millions)' represents the y-axis.

Step 2: Set Up the Graph

On the graphing calculator, set up a new graph. The x-axis will represent years since 1950, and the y-axis will represent the population in millions. Insert the data points from the table: (0, 2556), (10, 3039), (20, 3706), (30, 4453), (40, 5279), (50, 6083), and (60, 6849).

Step 3: Input the Exponential Equation

Enter the exponential equation \[ P(t) = 2686 e^{0.01604 t} \] into the graphing calculator. This equation models the population growth over time according to the exponential fit provided.

Step 4: Graph the Data and Equation

Use the graphing calculator to plot both the data points and the exponential curve simultaneously. Ensure the curve is set to be clearly visible, perhaps using a line, while points should remain distinct markers on the graph.

Step 5: Analyze the Graph

Examine the graph to see how well the exponential curve fits the population data points. Ideally, the curve should closely follow the trend of the data, showing how population growth is modeled by this exponential relationship.

Key Concepts

World Population

Understanding global population trends is crucial, as it affects resources, economies, and environments worldwide. The world population refers to the total number of humans living on Earth at any given time. This number has been steadily increasing since the 20th century, leading to significant discussions on sustainability and resource management.
By analyzing historical data, such as the table provided for the years since 1950, we can observe patterns and predict future changes. These predictions help policymakers and researchers to understand potential challenges and prepare accordingly. Population figures from specific decades, like those in the exercise, are used to model growth and track trends over time.

Curve Fitting

Curve fitting is a mathematical method used to find a curve that best represents a set of data points. In our exercise, we're interested in finding a curve that portrays how the world's population has grown over decades. This process helps in identifying trends and creating a smooth line that matches the data closely.
The exponential curve, specifically, is a powerful tool for population studies because it accounts for the nature of population growth, which tends to be faster as the population becomes larger. By understanding the best-fit curve, predictions for future population sizes can be made more accurately, which is crucial for planning resources and infrastructure.

Graphing Calculator

Graphing calculators are an essential tool in visualizing mathematical concepts like exponential growth. They allow us to plot data points and curves on a graph, providing a clear visual representation of abstract ideas. In the context of population data, a graphing calculator helps in plotting both real-world data points and the fitted exponential curve.
To use a graphing calculator effectively in this exercise:

• Set up your axes with 'Years since 1950' on the x-axis and 'Population (millions)' on the y-axis.
• Input the given data points.
• Enter the exponential equation provided.
• Plot the data and the curve simultaneously to observe their alignment.

The visual comparison on the calculator facilitates better understanding and illustrates how well the curve fits the historical data.

Population Modeling

Population modeling is essential in studying how populations grow and predict future changes. The model used in this exercise, an exponential model, takes the form \(P(t) = 2686 e^{0.01604 t}\). This equation uses two parameters:

• \(a = 2686\) which is the initial population close to the given value in 1950.
• \(b = 0.01604\) which determines the growth rate of the population.

Such models help demographers forecast future population sizes, revealing trends that depend on factors like growth rates and initial populations.
By applying these models, scientists and policymakers can anticipate population-related challenges and strategically plan for housing, resources, and services. Well-fitted models thus play a critical role in social and economic planning, ensuring sustainable development in the face of growing populations.