Problem 146 - College Algebra

Question

This group exercise involves exploring the way we grow. Group members should create a graph for the function that models the percentage of adult height attained by a boy who is \(x\) years old, \(f(x)=29+48.8 \log (x+1) .\) Let \(x=5,6\) \(7, \ldots, 15,\) find function values, and connect the resulting points with a smooth curve. Then create a graph for the function that models the percentage of adult height attained by a girl who is \(x\) years old, \(g(x)=62+35 \log (x-4)\) Let \(x=5,6,7, \ldots, 15,\) find function values, and connect the resulting points with a smooth curve. Group members should then discuss similarities and differences in the growth patterns for boys and girls based on the graphs.

Step-by-Step Solution

Answer

The calculated function values for boys and girls represent the percentage of adult height they have reached at different ages between 5 to 15. After plotting these values, it was observed that the growth rates differ between boys and girls, the specifics of which are subject to discussion and analysis. The precise differences and similarities can be better understood by scrutinizing the graph and discussing collectively.

Step 1: Compute Function Values for Boys

For the given age range \(x=5,6,7, \ldots, 15,\) substitute each value for \(x\) in the function \(f(x)=29+48.8 \log (x+1)\) and compute the corresponding function values.

Step 2: Plot the Points for Boys

Plot the age-height pairs obtained in Step 1 on a graph with age on the x-axis and height on the y-axis. Connect the points smoothly to represent the growth curve for boys.

Step 3: Compute Function Values for Girls

For the specified age range \(x=5,6,7, \ldots, 15,\) substitute each value for \(x\) in the function \(g(x)=62+35 \log (x-4)\) and find the corresponding function values.

Step 4: Plot the Points for Girls

Plot the age-height pairs obtained in Step 3 on the same graph used in Step 2. The age is on the x-axis and height is on the y-axis. Connect the resulting points smoothly to represent the growth curve for girls.

Step 5: Analyzing Patterns

Carefully inspect the plotted functions on the graph. Observe, compare, and discuss the trends and patterns between the growth patterns of both boys and girls.

Key Concepts

Logarithmic Functions

Logarithmic functions are a cornerstone in understanding growth patterns, especially because they can model situations where growth decreases as the value increases, which is common in real-life scenarios such as the rate of certain chemical reactions or even biological growth patterns.

For instance, the logarithmic function given in the exercise, represented as a model for growth is defined as the inverse of exponential functions and can be written in the form: \( f(x) = a + b \log (x+c) \). By analyzing these function models for growth, we observe that as \(x\) gets larger, the rate at which \( y \), or the growth, increases become gradually less. This reflects a natural slowdown in the rate of growth as a child approaches their adult height.

It’s important for students to become comfortable with the properties and graphs of logarithmic functions as it deepens their understanding of how certain variables evolve over time. When a student can grasp the impact of changing the constants—such as \(a\), \(b\), and \(c\) in the logarithmic function—they will have a better understanding of what the graph of this function represents in the real world.

Modeling Real-World Data

The utility of algebra becomes profoundly evident when we apply equations and functions to model real-world data. Growth patterns represent an excellent example of such applications since the progression of growth through time can be translated into mathematical models.

When students calculate the function values for both boys and girls by inputting the age values into the logarithmic functions and then graph these values, they are engaged in modeling. The smooth curves that connect the plotted points show a visual representation of the estimated progression of a child’s growth toward adult height. Modeling helps in understanding complex phenomena by simplifying them into equations that can be analyzed and manipulated, making them invaluable for predictions, planning, and interpretation.

Through this practice, students not only learn to apply logarithmic functions but also understand the broader concept of how algebra can be a powerful tool to make sense of the world around us. This blend of theory with empirical data is fundamental in fields such as biology, economics, and engineering.

Comparing Male and Female Growth

Biological growth patterns between males and females can be quite distinct, and algebraic functions allow us to compare these differences quantitatively. In the given exercise, different logarithmic models are used for boys and girls, indicating that these patterns do not follow the same exact path.

The significance of understanding how to compare male and female growth patterns using algebraic models goes beyond mere observation. It helps in identifying trends, pivotal points of growth, and the varying rates between genders. Analyzing these can offer insights into the nutritional, medical, and social needs during different stages of development.

Through mathematics, students are encouraged not only to calculate and plot but also to engage in a deeper discussion about how these patterns inform our understanding of gender-specific growth trends. This aspect of analysis cultivates critical thinking and a scientific approach to interpreting data, fostering a skill set that is relevant across a spectrum of disciplines.