Problem 108 - 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 \(x=1,2,3, \ldots, 12,\) find function values, and connect the resulting points with a smooth curve. Then create a function that models the percentage of adult height attained by a girl who is \(x\) years old, \(g(x)=\) \(62+35 \log (x-4) . \quad\) Let \(\quad x=5,6,7, \ldots, 15, \quad\) find function values, and connect the resulting points 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 exercise involves plotting two functions representing the growth patterns of boys and girls. Comparing these graphs helps observe the differences in growth rates between the boys and the girls.

Step 1: Calculate function values for boys

To find the percentage of height attained for a boy at each age from 1 to 12 years old, calculate \(f(x)=29+48.8 \log (x+1)\) for each value of \(x\).

Step 2: Plot function values for boys

Now, create a graph and plot the calculated percentage values for each corresponding age. Connect the dots to create a smooth curve that represents the growth of boys.

Step 3: Calculate function values for girls

Similarly, to find the percentage of height attained for a girl at each age from 5 to 15 years old, calculate \(g(x)=62+35 \log (x-4)\) for each value of \(x\).

Step 4: Plot function values for girls

On the same graph, plot the calculated percentage values for girls for each corresponding age. Connect these points to create a second curve that represents the growth of girls.

Step 5: Analysis and Comparison

Examine both graphs carefully and consider the rate and pattern of growth for boys and girls. Discuss similarities and differences in these patterns. Elevations, slopes, and curvatures of the graphs can provide insights into the growth changes over time.

Key Concepts

Logarithmic Functions

Logarithmic functions, such as the ones used in the group exercise to model human growth, play a crucial role in various fields for describing processes that change exponentially. A logarithmic function is generally expressed as
\[\begin{equation} f(x) = a + b \cdot \log(x - h) \end{equation}\]
where \(a\), \(b\), and \(h\) are constants, and \(x\) represents the independent variable. The base of the logarithm is often \(e\) (the natural logarithm) or 10, but it can also be adjusted to fit specific situations.

Understanding the Components on a Graph

When graphed, logarithmic functions exhibit a characteristic curve. The constant \(a\) vertically shifts the graph, while \(b\) affects the steepness. The \(h\) value horizontally shifts the graph. When \(b\) is positive, the graph increases, and when \(b\) is negative, the graph decreases. The crucial aspect of a logarithmic function is that it increases quickly at first and then gradually slows down, a pattern often observed in real-world situations such as radioactive decay, sound intensity, and indeed, human growth.

• Logarithmic growth is quick at the beginning and slows over time.
• The sign and value of constant \(b\) determine the growth direction and rate of the function.
• Comparing values of \(b\) among different logarithmic functions can hint at faster or slower growth rates.

The exercise provided demonstrates how logarithmic functions can model the complex nature of human growth and reflect different growth rates over ages. Through comparing the constants and evaluating the graphs, students can decipher real-life phenomena such as the varying growth patterns in boys and girls.

Modeling Growth with Functions

Functions are mathematical tools that model relationships between variables. Growth patterns are often described by functions in which the independent variable could be time, and the dependent variable is the quantity that grows.

Application to Real-life Phenomena

For example, in the textbook exercise, growth functions model the percentage of adult height attained by boys and girls over time. By plotting the values calculated from these functions, one can visualize the estimated physical growth during different periods of a person's life.

• Functions allow the prediction of future growth based on current trends.
• Different functions, logarithmic, linear, or exponential, are chosen based on the nature of growth.
• Real-life implications include agriculture, economics, and developmental biology, among others.

Through mathematical modeling, predictions and comparisons become possible, enabling informed decisions in diverse fields. Growth functions are an example of how abstract mathematical concepts are applied to understand the world in a quantifiable manner and provide insight into future developments.

Comparing Growth Rates

Comparing growth rates is fundamental to understanding the dynamics of different systems. Whether in biology, finance, or social sciences, growth rates can reveal how quickly changes occur over time and under various conditions.

Analysis Through Graphs

Graphs serve as visual representations for comparing the growth rates of different functions. As seen in the group exercise, examining the curves' elevations, slopes, and overall shapes provides insights into how rapidly or slowly the subjects represented by these functions are changing.

• Slope of the curve can indicate the rate of growth at any point.
• Inflection points on a graph highlight where growth rates change substantially.
• Intersection points can mark where two different growth processes equalize.

By comparing those graphical features in the boys' and girls' growth models, one can discern not only the similarities and differences in height growth patterns but also gauge the age at which growth rates alter. This comparison illuminates how various factors can influence growth rates differently, leading to a deeper understanding of the underlying processes.