Problem 104 - Elementary and Intermediate Algebra

Question

The function \(P(a)=41.0+20.4 \ln a\) approximates the percent of adult height attained by an earlymaturing girl of age \(a\) years, for \(1 \leq a \leq 18 .\) The function \(P(a)=37.5+20.2 \ln a\) does the same for a late-maturing girl. Find the difference in percent of their adult height for both maturity types on their 10 th birthday. (Source: Growth, Maturation, and Physical Activity, Human Kinetic Books, Robert Malina)

Step-by-Step Solution

Answer

The early-maturing girl is about 4% taller than the late-maturing girl on their 10th birthday.

Step 1: Identify the Functions

The problem gives us two functions to approximate the percent of adult height attained, one for early-maturing girls and one for late-maturing girls:- Early-maturing girl: \( P_e(a) = 41.0 + 20.4 \ln a \)- Late-maturing girl: \( P_l(a) = 37.5 + 20.2 \ln a \).

Step 2: Substitute Age into the Functions

Substitute \( a = 10 \) into both functions to find the percent of adult height attained by both types of maturing girls on their 10th birthday.

Step 3: Calculate for Early-Maturing Girl

Calculate \( P_e(10) \):\[ P_e(10) = 41.0 + 20.4 \ln(10) \]Using the approximation \( \ln(10) \approx 2.302 \), we get:\[ P_e(10) = 41.0 + 20.4 \times 2.302 \]\[ P_e(10) \approx 41.0 + 46.9608 \]\[ P_e(10) \approx 87.9608 \]

Step 4: Calculate for Late-Maturing Girl

Calculate \( P_l(10) \):\[ P_l(10) = 37.5 + 20.2 \ln(10) \]Using the same approximation \( \ln(10) \approx 2.302 \), we get:\[ P_l(10) = 37.5 + 20.2 \times 2.302 \]\[ P_l(10) \approx 37.5 + 46.4604 \]\[ P_l(10) \approx 83.9604 \]

Step 5: Find the Difference in Percent

Subtract the percent of adult height for the late-maturing girl from the early-maturing girl on their 10th birthday:\[ \Delta P = P_e(10) - P_l(10) \]\[ \Delta P = 87.9608 - 83.9604 \]\[ \Delta P \approx 4.0004 \]

Key Concepts

Functions

In mathematics, a function is a special relation between a set of inputs and a set of permissible outputs. In simple terms, each input is related to exactly one output. Functions are like machines: you feed an input into the machine, it does something to it (like performing a calculation), and spits out an output. This concept is critical because it forms the basis for expressing and analyzing many mathematical relationships.
For the given problem, we deal with two specific functions, each representing a different scenario of growth.

• The function for early-maturing girls is denoted as \(P_e(a) = 41.0 + 20.4 \ln a\).
• The function for late-maturing girls is \(P_l(a) = 37.5 + 20.2 \ln a\).

These functions take an age as input and provide the corresponding percent of adult height attained as output. Understanding functions helps us predict outcomes based on different scenarios by plugging different input values into our function formulae.

Logarithms

Logarithms are a way of expressing the power to which a number must be raised to obtain another number. For example, the logarithm of 100 to base 10 is 2 because 10 squared equals 100. The notation \(\ln\) refers to the natural logarithm, which uses the constant \(e\) (approximately 2.71828), as its base.
In our problem, we see logarithms used in modeling growth over age:

• In \(P_e(a) = 41.0 + 20.4 \ln a\), the term \(\ln a\) modifies how quickly height is attained.
• Similarly, \(P_l(a) = 37.5 + 20.2 \ln a\) manipulates the growth prediction based on age.

Logarithms are particularly useful in modeling growth because they can capture the diminishing rate of increase over time, which often mirrors real-world growth patterns such as height.

Problem Solving

Problem solving in mathematics often involves a structured approach, using a series of steps to break down and tackle the problem. By following clear steps, we can ensure that all parts of a problem are addressed logically and efficiently.
In this exercise, we used the following steps:

• Identify and understand the functions involved, noticing what each represents for early and late-maturing girls.
• Substitute the specific input (age) into each function to find the specific outputs or percent of height attained.
• Perform the calculations needed to interpret these outputs at age 10.
• Finally, compare the outputs to determine the difference in height attainment between both groups.

Approaching math problems with a methodical, step-by-step approach can make seemingly complex problems much more manageable.

Mathematical Modeling

Mathematical modeling is a technique that uses mathematical structures and relationships to represent and analyze real-world phenomena. In our exercise, mathematical modeling is used to project the height growth of girls over time based on their maturity type.
Creating a model involves proposals for when and why certain changes occur, often based on observed data and patterns. For example, our models for early-maturing and late-maturing girls incorporate age and logarithmic functions tuned to past data to estimate growth:

• Early maturity: \(P_e(a) = 41.0 + 20.4 \ln a\).
• Late maturity: \(P_l(a) = 37.5 + 20.2 \ln a\).

These models allow for predictions not just at one point but across a range of ages, helping to explore complex healthcare questions like height development in adolescence. Mathematical modeling, therefore, provides tools for theoretical forecasts well beyond direct observation, making it invaluable in fields ranging from biology to economics.