Problem 73 - College Algebra and Calculus: An Applied Approach

Question

Body Mass Index Body mass index (BMI) is a measure of body fat based on height and weight. The 75 th percentile BMI for females, ages 9 to 20 , grew more slowly than that of males of the same age range. Models that represent the 75 th percentile BMI for males and females, ages 9 to 20 , are given by \(\left\\{\begin{array}{ll}B=0.73 a+11 & \text { Males } \\ B=0.61 a+12.8 & \text { Females }\end{array}\right.\) where \(B\) is the \(\mathrm{BMI}\left(\mathrm{kg} / \mathrm{m}^{2}\right)\) and \(a\) represents the age, with \(a=9\) corresponding to 9 years old. Use a graphing utility to determine whether the BMI for males will exceed the BMI for females. (Source: National Center for Health Statistics)

Step-by-Step Solution

Answer

The graph becomes a visual representation of the BMI distribution for males and females. If at any point within the age range, the line for males exceeds the line for females, that will mean males have greater BMI than females. The exact point of intersection or overtaking can be ascertained through a precise graphing tool.

Step 1: Identify model equations

The given models are: \(B = 0.73a + 11\) for males and \(B = 0.61a + 12.8\) for females, where B is the Body Mass Index (BMI) and a represents the age.

Step 2: Using Graphing utility

First load the graphing utility program for your computer. Now, plot these two equations on a graph where the x-axis represents 'a' (age) and the y-axis represents 'B' (BMI).

Step 3: Examine the Graph

Look at how the two lines intersect each other. If the male line exceeds the female line on the graph at any point within the age range of 9 to 20, then it means that the BMI for males exceeds that of females at that particular age.

Key Concepts

BMI Modeling

Body Mass Index (BMI) is a widely used indicator for assessing body fat in relation to an individual's weight and height. In the educational context of BMI modeling, this concept involves creating mathematical representations to describe changes in BMI over time or across different variables, such as age.

The exercise provided models for the 75th percentile BMI for both males and females between the ages of 9 to 20. These models are linear equations, with each having a slope and an intercept. The slope represents the rate of increase in BMI with each year of age, while the intercept indicates the estimated BMI at the age of 9.

Importantly, the models are derived from statistical analysis of population data and are useful for making predictions or comparative analysis. For example, by comparing the two equations—\[\[\begin{align*}B &= 0.73a + 11 \ B &= 0.61a + 12.8\text{,}\end{align*}\]\]where B is BMI and a is age—we can explore patterns like whether males generally have higher BMI than females at the same age.

It's vital for students to realize that these models are simplifications and should not substitute for medical advice. BMI modeling helps in grasping larger population health trends and informing public health decisions.

Graphing Utility Application

Graphing utilities, such as graphing calculators or computer software, are powerful tools for visualizing relationships and patterns in data. This exercise employed such a utility to visualize the relationship between age and BMI for males and females.

By entering the linear equations into a graphing utility, students can quickly see where and how the lines for males and females intersect or diverge. The utility effectively translates abstract formulas into a visual graph that illustrates the BMI trends as age increases.

To help students fully grasp this concept, it’s beneficial to guide them through the process of inputting the equations, setting an appropriate scale for the axes, and interpreting the resulting graph. Encourage students to take note of the slope and y-intercept visually, understanding that each graph line represents the trend of BMI changes in the given demographic.

For instance, when looking at the intersection point, if any, students can deduce the age at which BMI is equal for both genders and how it changes afterward. This hands-on approach using a graphing utility consolidates their understanding of both the math involved and the practical implications of data trends.

Comparative Statistical Analysis

Comparative statistical analysis involves evaluating data sets to discern patterns, differences, and similarities. In the BMI exercise, this type of analysis is key to understanding how BMI changes for different genders over a range of ages.

Through comparative statistical analysis, students can identify trends within the models for males and females. The coefficients in the linear equations (slopes of 0.73 for males and 0.61 for females) provide insight into the rate at which BMI increases with age. By comparing these rates, students can detect that, according to the models, BMI increases more rapidly for males than for females during the ages under consideration.

Furthermore, a comparative analysis includes examining the y-intercepts (11 for males and 12.8 for females), indicating that females start with a higher estimated BMI at age 9. This type of investigation can lead to discussions about various factors that contribute to changes in BMI, enhancing students' ability to critically analyze statistical data.

Understanding how to conduct and interpret comparative statistical analysis equips students with valuable skills for their studies and beyond, allowing them to make informed judgments about data related to health, economics, social sciences, and more.