Problem 68 - A Graphical Approach to Precalculus with Limits

Question

The average weight for a man can sometimes be estimated by $$f(x)=0.117 x^{1.7}$$ where \(x\) represents the man's height in inches and \(f(x)\) is his weight in pounds. What is the average weight of a 68 -inch-tall man?

Step-by-Step Solution

Answer

The average weight of a 68-inch-tall man is approximately 81.81 pounds.

Step 1: Identify the Function and Variables

The given function is \( f(x) = 0.117 x^{1.7} \) where \( x \) is the height in inches, and \( f(x) \) is the weight in pounds. The task is to find the average weight of a man with height \( x = 68 \) inches.

Step 2: Substitute the Height into the Function

Take the height of the man, which is 68 inches, and substitute it into the function. We have:\[ f(68) = 0.117 imes 68^{1.7} \]

Step 3: Calculate the Power

Calculate \( 68^{1.7} \). Using a calculator, this becomes approximately \( 68^{1.7} \approx 697.62 \).

Step 4: Apply the Coefficient

Multiply the result by the coefficient 0.117:\[ 0.117 imes 697.62 \approx 81.81 \]

Step 5: Interpret the Result

The computed result \( 81.81 \) represents the average weight in pounds of a 68-inch-tall man.

Key Concepts

Mathematical Modeling

Mathematical modeling is a powerful tool used to represent real-world scenarios with mathematical expressions. It simplifies complex real phenomena into manageable formulas or functions, enabling predictions and analysis. In the exercise, the average weight of a man is estimated using the function \( f(x) = 0.117 x^{1.7} \). Here, the function models the relationship between a man's height, \( x \), and his average weight, \( f(x) \), in pounds. This type of modeling is beneficial as it offers a quick and generally accurate method to estimate someone's weight based on height.

Such models are created by observing patterns and relationships in data, helping us make reasonable estimates. They help in fields like health, physics, and economics, where understanding and predicting behaviors are crucial. However, while they can provide insights, it is essential to remember that models are simplifications and may not account for all variables involved in a real-life situation.

Exponents and Powers

Exponents and powers are mathematical operations that involve raising numbers to specified levels. When using functions like \( f(x) = 0.117 x^{1.7} \), understanding exponents is key. An exponent indicates the number of times a number (known as the base) is multiplied by itself.

In this formula, the exponent is 1.7, suggesting that we raise the height \( x \) (in this case, 68 inches) to the power of 1.7. This operation doesn't mean to multiply 68 by itself 1.7 times, but rather it uses logarithmic principles to find a precise value between whole number exponents.

This concept is crucial because it allows the function to better fit the data it is modeling. By choosing specific exponents, the model reflects patterns or relationships found in empirical data more accurately, improving its usability in real-world predictions.

Substitution in Functions

Substitution in functions is a simple but fundamental concept that involves replacing a variable with a given value. In the given task, we substitute \( x = 68 \) into the function \( f(x) = 0.117 x^{1.7} \).

• Begin by identifying the function and the value to substitute.
• Replace each instance of \( x \) with the given value: here, 68.
• Calculate the result of the function with this substitution.

This step-by-step process translates the model's general rule into a specific scenario, allowing us to find the expected outcome (the man's average weight). Substituting values help in testing models and making predictions or deductions based on particular details.

Substitution is a widely-used technique in many mathematical calculations, enabling users to manage variables and constants with ease. It is particularly useful in mathematical modeling, facilitating the application of general models to individual cases.