Problem 60 - Precalculus with Calculus Previews

Question

A mathematical model for estimating body surface area \(S\) (in square meters) is given by $$ \log _{10} S=-0.69364+(0.425) \log _{10} w+(0.725) \log _{10} h, $$ where \(w\) and \(h\) are a person's weight (in kilograms) and height (in meters), respectively. This empirical formula, due to D. Dubois and E. F. Dubois, first published in the Archives of Internal Medicine in 1916 , is still used today by medical researchers. (a) Use (14) to estimate the body surface area of a person whose weight is \(w=70 \mathrm{~kg}\) and who is \(h=1.75 \mathrm{~m}\) tall. (b) Determine your weight and height and estimate your own body surface area.

Step-by-Step Solution

Answer

The body surface area is approximately 1.85 square meters.

Step 1: Plug in Given Values

We have the formula \( \log_{10} S = -0.69364 + (0.425) \log_{10} w + (0.725) \log_{10} h \). To estimate the body surface area for a person with weight \( w = 70 \) kg and height \( h = 1.75 \) m, substitute these values into the formula.

Step 2: Calculate Logarithms

Calculate \( \log_{10} 70 \) and \( \log_{10} 1.75 \). Using logarithm tables or a calculator, we find: \( \log_{10} 70 \approx 1.845 \) and \( \log_{10} 1.75 \approx 0.243 \).

Step 3: Substitute Logarithms into Formula

Substitute the calculated logarithms into the body surface area formula: \[ \log_{10} S = -0.69364 + (0.425)(1.845) + (0.725)(0.243) \].

Step 4: Evaluate the Right-Hand Side

Compute each part of the equation: \( (0.425)(1.845) = 0.784125 \) and \( (0.725)(0.243) = 0.176175 \). Add these results to \(-0.69364\): \[ \log_{10} S = -0.69364 + 0.784125 + 0.176175 = 0.26666 \].

Step 5: Convert from Logarithm to Linear Scale

To find \( S \), take the antilogarithm (base 10) of 0.26666: \( S = 10^{0.26666} \). This results in \( S \approx 1.85 \) square meters.

Step 6: Estimate Personal Body Surface Area

To find your own body surface area, use your personal values for weight \( w \) and height \( h \). Follow the same steps: calculate the logarithms of your weight and height, substitute them into the formula, evaluate, and take the antilogarithm to find \( S \).

Key Concepts

Logarithmic Functions

Logarithmic functions play a crucial role in making complex calculations more straightforward. When we use logarithms, we're essentially dealing with exponents. In this context, the expression \( \log_{10} S = -0.69364 + (0.425) \log_{10} w + (0.725) \log_{10} h \) is used for scaling purposes.
It allows us to work with large or small numbers by converting multiplication into addition. This simplifies the process of estimating body surface area which involves both weight and height.
In practical terms, the logarithmic expression is helpful because:

• Adding logarithms is more manageable than multiplying the raw values of height and weight directly.
• Logarithms help in analyzing data patterns, especially in physiological attributes, which often involve exponential growth.
• Using base 10 logarithms aligns well with our decimal-based numbering system.

To interpret \( \log_{10} S \) effectively, remember, it's a representation of an exponent—that is, what power you need to raise 10 to get \( S \). Finally, after calculations, you take the antilogarithm to return \( S \) to its original scale.

Empirical Formulas

Empirical formulas are derived from experimental data and are used to approximate real-world scenarios. While they may not be perfect, they provide us with an efficient way to estimate variables such as body surface area.
The formula \( \log_{10} S = -0.69364 + (0.425) \log_{10} w + (0.725) \log_{10} h \) serves as an empirical method to calculate body surface area based on patterns observed by Dubois and Dubois.
Empirical formulas are beneficial because:

• They offer close approximations derived from large-scale observation data.
• They have been tested over time, ensuring a certain level of accuracy and reliability.
• These formulas save time and resources compared to more complex and potentially costly models.

This specific formula balances simplicity with practical accuracy, providing medical professionals and researchers an easy-to-use tool for quick estimates.

Mathematical Modeling

Mathematical modeling is the process of using mathematical expressions to represent real-world phenomena. This involves the crafting of a function or set of equations that can replicate the nuances of physical aspects.
In our case, the model \( \log_{10} S = -0.69364 + (0.425) \log_{10} w + (0.725) \log_{10} h \) models how weight and height relate to body surface area.
Mathematical models are crucial because they:

• Provide a framework to understand complex systems through simpler, abstract representations.
• Enable predictions and hypothesis testing in controlled or variable environments.
• Replicate interdependencies in a model, such as how changes in weight or height can affect body surface area.

This particular model highlights the proportional contributions of height and weight through coefficients 0.425 and 0.725, illustrating the sophisticated relationship these variables have on body surface area estimation. Through such models, we can simulate, analyze, and optimize the parameters of interest efficiently.