Problem 82 - College Algebra

Question

The percentage of adult height attained by a girl who is \(x\) years old can be modeled by $$f(x)=62+35 \log (x-4)$$ where \(x\) represents the girl's age (from 5 to 15 ) and \(f(x)\) represents the percentage of her adult height. Use the function to solve Exercises. Approximately what percent of her adult height is a girl at age ten?

Step-by-Step Solution

Answer

To calculate the exact percentage, use a calculator to find the value of \( \log(6) \), multiply it with 35 and add 62. This will provide the percentage of the girl's adult height at age 10.

Step 1: Understanding the Equation

This problem uses the function \(f(x)=62+35 \log (x-4)\). Here, \(x\) is the age of the girl and \(f(x)\) is the percentage of her adult height. Determine the percentage of adult height when the girl is 10 years old, substitute \(x=10\) in the function.

Step 2: Substitution

Replace \(x\) with 10 in the function: \(f(10)=62+35 \log (10-4)=62+35 \log (6)\)

Step 3: Calculation

Calculate the value of \(f(10)=62+35 \log (6)\) to get the percentage of her adult height at age 10.

Key Concepts

Algebraic Modeling

Algebraic modeling is the process of representing real-world situations through mathematical models using algebraic expressions or equations. In algebraic modeling, variables and constants are used to mimic behaviors observed in practical contexts, allowing predictions or analyses of different scenarios.

For instance, the function given in the exercise, \(f(x)=62+35 \log(x-4)\), is an algebraic model. This specific function represents the growth pattern of a girl’s height relative to her age. Through algebraic modeling, we can estimate the percentage of a girl's adult height at a certain age (which in this scenario is between 5 to 15 years old). The beauty of algebraic modeling lies in its ability to transform observable data into a formula that can be manipulated to forecast outcomes under various conditions.

Function Substitution

Function substitution is a technique used to evaluate functions at specific input values. It involves replacing the function's variable with a concrete value to compute the function's output. This method is particularly useful in understanding the behavior of functions and making specific calculations.

From our exercise, the function substitution comes into play when we insert the age of the girl, 10, into the variable \(x\) in the function \(f(x)\). This is shown in the step-by-step solution, where \(x\) is replaced with 10, resulting in \(f(10)=62+35 \log(10-4)\). By this action, we're able to determine the percentage of adult height the model predicts for a 10-year old girl.

Logarithms

Logarithms are the inverses of exponents; they help us determine the power to which a number (the base) must be raised to produce another number. In the mathematical expression \(y = \log_b(x)\), \(y\) is the logarithm of \(x\) to the base \(b\), and it answers the question: to what power must we raise \(b\) to obtain \(x\)?

In the context of the given problem, we see the logarithm appearing within the growth model as \(35 \log(x-4)\). The logarithmic function here, with its base 10 implied, illustrates how the percentage of adult height increases as the girl grows older. Understanding how to work with logarithms, including calculating and interpreting them, is critical to solving algebraic problems with growth models, as it transforms multiplicative relationships into additive ones, which are often easier to analyze.

Mathematical Functions

Mathematical functions are relations that uniquely associate members of one set, which are known as inputs or arguments, with members of another set, which are the outputs or function values. In essence, every input is assigned exactly one output. Functions are usually written in the form \(f(x)\) where \(x\) is the input value and \(f(x)\) is the resulting output.

The function provided in the exercise, \(f(x)=62+35 \log(x-4)\), can be used to calculate various outcomes across the range of the domain, which in this case is limited to ages 5 through 15. Functions like this help in creating a predictable pattern for evaluation and make the study of changes in relationships systematic and quantifiable. They are fundamental in various fields of science and mathematics due to their universal applicability and the clarity they provide in modeling situations.