Question

For the next set of exercises, use the following table, which features the world population by decade. $$ \begin{array}{|l|l|} \hline \text { Years since 1950 } & \text { Population (millions) } \\ \hline 0 & 2,556 \\ \hline 10 & 3,039 \\ \hline 20 & 3,706 \\ \hline 30 & 4,453 \\ \hline 40 & 5,279 \\ \hline 50 & 6,083 \\ \hline 60 & 6,849 \\ \hline \end{array} $$ 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.

Short answer

Graph the scatter plot and curve on the calculator to compare observed data with the exponential model.

Step-by-step solution

Understand the Data Table and Function

The table provided shows the world population in millions against the number of years since 1950. The exponential function given is \( P(t) = 2686 \cdot e^{0.01604t} \). This function aims to model the population data.

Set Up the Graphing Calculator

Turn on your graphing calculator and enter the table data as a scatter plot. Input the years since 1950 as the independent variable \( x \) and the population (in millions) as the dependent variable \( y \).

Enter the Exponential Function

Enter the provided best-fit exponential curve \( P(t) = 2686 \cdot e^{0.01604t} \) in the function editor of the calculator. This needs to be plotted on the same graph as the scatter plot data.

Adjust Window Settings

Adjust the window settings of your graphing calculator to appropriately scale the x-axis and y-axis so that both the data points and the curve are visible. The x-axis could range from 0 to 60, and the y-axis should cover at least from 2500 to 7000.

Graph the Data and Function

With the data inputted and the function set, draw the scatter plot and the exponential function on the same graph. This provides a visual comparison between the measured data and the exponential model.

Evaluate the Fit

Examine the graph to see how well the exponential curve fits the data. A good fit will have the curve closely follow the scatterplot points, indicating the function is a suitable model for the data.

Key concepts

Graphing Calculator

A graphing calculator is a powerful tool for visualizing mathematical functions and data comparisons. To graph the world population data, you should first enter the scatter plot data into the calculator.
This involves inputting the years since 1950 as your x-values and the corresponding population figures as y-values. After setting up the scatter plot, switch to the function editor on your graphing calculator. Here, you can enter the given exponential function, like \( P(t) = 2686 \cdot e^{0.01604t} \), which models the population growth.
Make sure this curve plots alongside your scatter plot.Setting the graph window appropriately is vital for a clear view of both the data points and the curve. It typically involves adjusting the x-axis range from 0 to 60 years and the y-axis to encompass 2500 to 7000 (millions), accommodating all data and the curve. With these steps, you can visualize the alignment of your data points with the model function.

World Population Data

World population data gives insights into how the number of people on the planet changes over time. In this exercise, you start from the year 0, which reflects 1950, and progress in ten-year intervals up to 60 years.
For example, year 0 equates to 2,556 million people, and year 60 corresponds to 6,849 million people. This sort of dataset is crucial for identifying trends and patterns, helping us understand how the population grows or decreases.
In this case, you see an upward trend, showing continuous population growth over each decade. This growth can be efficiently modeled with an exponential curve due to its consistent rate of increase each period.
Utilizing such data over long periods allows demographers and policymakers to predict future growth and plan accordingly, impacting global strategies in healthcare, education, and resource management.

Best-Fit Curve

A best-fit curve is a mathematical function that best represents a set of data points. The goal is to find a curve that closely aligns with the data, visually and mathematically.
In the context of exponential growth, a best-fit curve like \( P(t) = 2686 \cdot e^{0.01604t} \) models the world population data well.This specific exponential function was selected by analyzing the data points and determining the parameters that make the function's path closely follow these points. The constant 2686 represents the initial amount, while \( 0.01604 \) is the growth rate. Both components contribute to how the curve fits the data.
Using a best-fit curve is beneficial because it offers a simplified equation that captures the key characteristics of the data. You can apply this model to forecast future trends, within reasonable limits. It provides demographers a tool for illustrating large-scale trends like world population growth, offering a clear picture for informed decision-making.