Problem 148 - Algebra and Trigonometry Real Mathematics, Real People

Question

The percent \(m\) of American males between the ages of 18 and 24 who are no more than \(x\) inches tall is modeled by $$m(x)=\frac{100}{1+e^{-0.6114(x-69.71)}}$$ and the percent \(f\) of American females between the ages of 18 and 24 who are no more than \(x\) inches tall is modeled by $$f(x)=\frac{100}{1+e^{-0.66607(x-64.51)}}$$ (Source: U.S. National Center for Health Statistics) (a) Use a graphing utility to graph the two functions in the same viewing window. (b) Use the graphs in part (a) to determine the horizontal asymptotes of the functions. Interpret their meanings in the context of the problem. (c) What is the average height for each sex?

Step-by-Step Solution

Answer

The horizontal asymptotes of the functions are at \(m(x) = 100%\) and \(f(x) = 100%\) respectively, indicating that no matter how tall a person is, the percentage of people that height or shorter will not exceed 100%. The average height for males is 69.71 inches (about 5 feet 10 inches) and for females is 64.51 inches (about 5 feet 5 inches)

Step 1: Graph the Functions

To begin, use a graphing utility to plot the two functions \(m(x)=\frac{100}{1+e^{-0.6114(x-69.71)}}\) and \(f(x)=\frac{100}{1+e^{-0.66607(x-64.51)}}\) on the same graph. You should see two distinct curves.

Step 2: Identify the Horizontal Asymptotes

An asymptote is a line that the graph of a function approaches but never touches. By analyzing the two plotted functions, you can see that both functions level off and approach a horizontal line, indicating a horizontal asymptote. Mathematically you can determine this because as \(x\) goes to infinity, the exponential term \(e^{-0.6114(x-69.71)}\) and \(e^{-0.66607(x-64.51)}\) approaches 0, leaving just the numerator 100.

Step 3: Interpret the Asymptotes

In the context of this exercise, these asymptotes mean that no matter how tall people get, the percentage of the population that is that height or shorter will never exceed 100%.

Step 4: Determine the Average Heights

To find the average height for each sex, set the inside of the exponential equal to 0 and solve for \(x\). For males, you get \(x = 69.71\) inches (about 5 feet 10 inches), and for females, you get \(x = 64.51\) inches (about 5 feet 5 inches). This means that the average American male between 18 and 24 years old is around 5 feet 10 inches tall, and the average female is around 5 feet 5 inches tall.

Key Concepts

Graphing Functions

Graphing functions is a valuable skill in mathematical modeling. This exercise involves graphing two functions that model the heights of American males and females between ages 18 to 24. Using a graphing utility, these functions are plotted to visually represent the distribution of heights.

• The function for males is given by \(m(x)=\frac{100}{1+e^{-0.6114(x-69.71)}}\).
• The function for females is given by \(f(x)=\frac{100}{1+e^{-0.66607(x-64.51)}}\).

The graphs of these functions typically produce sigmoid curves. These curves will also reveal the proportion of the population shorter than a certain height.

When plotting, observe: as the value of \(x\) (height) increases, both curves approach a horizontal line. This represents a limit or threshold that is not surpassed. Thus, by visually analyzing these functions, you can gain insights into population height distributions.

Exponential Functions

Exponential functions are present in this exercise to model the cumulative distribution of heights. In both functions, the exponential term includes an expression that modifies height (\(x\)). The general form looks like this:

• For males: \(e^{-0.6114(x-69.71)}\)
• For females: \(e^{-0.66607(x-64.51)}\)

In these expressions, the coefficient determines the rate at which the function approaches its asymptote. An important characteristic of exponential functions is their behavior as \(x\) increases: the exponential decay leads these functions to approach certain constant values, or asymptotes.

The role of the exponential component in this context is to adjust how quickly the percentage function levels out to 100. This tailing off to a maximum reflects the biological limit on how tall the population can be.

Average Height Analysis

Determining average height is essential in interpreting data from these models. In particular, this task uses the turning point of the function where the exponential term equals zero:

For American males, setting the inside of the exponential term \(-0.6114(x-69.71) = 0\) results in an average height of \(x = 69.71\) inches (around 5 feet 10 inches).

• Equation: \(-0.6114(x-69.71) = 0\)
• Solves to: \(x = 69.71\)

For American females, resolving \(-0.66607(x-64.51) = 0\) gives an average height of \(x = 64.51\) inches (approximately 5 feet 5 inches).

• Equation: \(-0.66607(x-64.51) = 0\)
• Solves to: \(x = 64.51\)

These values provide straightforward measures to understand the central tendency in height data for young adults in the U.S. It emphasizes how algebraic modeling can inform us about real-world data distributions.