91Ó°ÊÓ

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..... \(15,\) 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

01

Calculate function values for boys

First, find the values of the function \(f(x)=29+48.8 \log (x+1)\) for each year from 5 to 15. These values will represent the percentage of adult height attained by a boy at these ages.

02

Plot function values for boys

Using the function values computed in step 1, plot the points on a graph. The age should be represented on the x-axis and the computed function value on the y-axis. Connect the points with a smooth curve.

03

Calculate function values for girls

Now find the values of the function \(g(x)=62+35 \log (x-4)\) for each year from 5 to 15. These values will show the percentage of adult height attained by a girl at these ages.

04

Plot function values for girls

Like in step 2, plot the points calculated in step 3. Use the same x and y axes as before. Connect the points with a smooth curve.

05

Analyze the graphs

Once both the graphs are plotted on the same coordinate axis, analyze the differences and similarities in their growth patterns. Discuss these findings within the group.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graphing Functions

When graphing functions, it helps to visualize data points and how they relate to each other. In our given exercise, we have two functions that model the growth patterns of boys and girls. For boys, the function is given by: \( f(x) = 29 + 48.8 \log (x+1) \) This equation models the percentage of adult height a boy reaches at age \( x \). For girls, the function is: \( g(x) = 62 + 35 \log (x-4) \) This equation represents the percentage of adult height attained by a girl at age \( x \).
To graph these functions:

• Determine and plot each function value for ages 5 to 15.
• Connect these points smoothly to demonstrate growth patterns.

By doing so, we can easily visualize and compare how the growth progresses with age for both genders.

Logarithmic Functions

Logarithmic functions, like the ones used in our models, are great tools for representing phenomena like growth. They help us understand the rate of change for different processes.

• The general form of a logarithmic function is: \( y = a + b \log(x) \).
• In our equations, \( f(x) \) and \( g(x) \) are examples of such functions adjusted by constants \( a \) and coefficients \( b \) to fit specific growth data.

Logarithms are particularly effective in growth modeling because they can describe rapid growth rates initially, which then slow down as the subject approaches its maximum, or adult height in this context. This aligns well with the physiological growth patterns seen in human development.

Gender Differences in Growth

The exercise highlights the use of functions to illustrate gender differences in growth. Boys and girls grow at different rates and reach their adult heights through unique patterns.

• Boys tend to have a more consistent growth rate as modeled by \( f(x) \), indicating steady progress over the years.
• Girls, shown by \( g(x) \), exhibit different growth dynamics, often starting earlier and reaching a plateau sooner than boys.

By analyzing the curves produced from these functions, one can see that these differences are significant and can be boxed into distinct stages of growth. Such insights are important for understanding broader biological factors that influence development.

Function Plotting

Plotting functions involves a few key steps that help us clearly represent and interpret our data visually. Here’s how to approach it using our exercise as an example:

• Calculate the function's output for each input value (ages 5 to 15).
• Plot these calculated values on a coordinate plane where the x-axis represents age and the y-axis represents the percentage of adult height.
• Once plotted, connect these points with a smooth line to reveal the growth curves clearly.

Function plotting reveals important trends and differences in growth patterns for boys and girls. The visual representation makes it easier to discuss such trends among peers and teachers, providing a tangible view of abstract math concepts.