Problem 81 - 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 \(13 ?\)

Step-by-Step Solution

Answer

The answer should be the calculated value of \(f(13)\), which represents the percentage of adult height the girl has reached at age 13. Prepare a calculator for detailed calculation and solve for the result.

Step 1: Define the Problem

We are given the function \(f(x)=62 + 35 \log (x - 4)\) which models the height of a girl at age \(x\). We need to find the value of \(f(x)\) when \(x=13\). This will give us the percentage of her adult height the girl has reached at age 13.

Step 2: Substitute the Value

We substitute \(x=13\) into our formula. So, our function becomes \(f(13) = 62 + 35 \log (13-4)\).

Step 3: Calculate Logarithm and Solve

Our function now contains a logarithm, \(\log(13-4)\). Subtracting 4 from 13 gives us 9, so we calculate the logarithm of 9. Then multiply the result with 35 and add 62 to get our final result for \(f(13)\). This process would be much easier with a calculator handy.

Key Concepts

Logarithmic Functions

Logarithmic functions are an essential part of algebraic functions used for modeling real-world scenarios. In the context of the given exercise, the logarithmic function is used to estimate the percentage of adult height a girl has reached at a specific age. Logarithms are the inverse operation of exponentiation which means they help us determine the power to which a number must be raised to obtain another number.
For example, in the problem, the function is given as:

• \( f(x) = 62 + 35 \log(x - 4) \)

The expression \( \log(x - 4) \) denotes the logarithm of the expression \((x - 4)\). In simpler terms, a logarithm answers the question: "To what exponent must 10 be raised, to yield \(x - 4\)?"
In our model, \(35 \log(x - 4)\) represents the variable component scaling the impact of age on height, while the constant 62 represents the baseline height percentage. Understanding how each component affects the function helps in predicting phenomena linked to growth and time. Always use calculators for precise calculations when dealing with logarithmic evaluations.

Modeling with Mathematics

Mathematical modeling is an integral concept in converting real-life situations into mathematical expressions. In our exercise, the task is to model the growth of a girl's height using a specific formula. Such models provide a simplified way to predict and analyze complex systems by using mathematical structures.
In the given function, \(f(x) = 62 + 35 \log(x - 4)\), the height percentage is determined by connecting age with height growth. Here are the key points about this model:

• The term "62" acts as a constant baseline percentage for the height.
• "35 \log(x - 4)" represents the growth factor that changes logarithmically with age.
• This model is valid only within the age range of 5 to 15 years, considering a plausible age for calculating human growth percentage.

Using such mathematical models helps in efficiently dealing with predictions using existing data, guiding researchers and decision-makers in many domains like biology, economics, and even engineering.

Substitution Method

The substitution method is a convenient algebraic technique for solving equations. It involves replacing a variable with a particular value to simplify and solve the function for that specific input. This method is extremely handy in situations like ours, where you need to find the output of a function at a specific value.
In the current exercise, the substitution process is straightforward:

• Set the value of \( x = 13 \) in the function \( f(x) = 62 + 35 \log(x - 4) \).
• Substitute to simplify the expression to \( f(13) = 62 + 35 \log(13 - 4) \).
This simplifies to \( f(13) = 62 + 35 \log(9) \). You then calculate \( \log(9) \) using a calculator and multiply by 35 before finally adding 62.
This substitution and calculation will give the percentage of adult height a girl reaches at age 13. This method provides a stepwise approach to problem-solving and is especially valuable when dealing with complex functions.