Problem 135 - College Algebra with Modeling and Visualization

Question

The surface area of the skin covering the human body is a function of more than one variable. A taller person tends to have a larger surface area, as does a heavier person. Both height and weight influence the surface area of a person's body. A formula to determine the surface area of a person's body in square meters is given by \(S(w, h)=0.007184 w^{0.425} h^{0.725},\) where \(w\) is weight in kilograms and \(h\) is height in centimeters. Use \(S\) to estimate the surface area of a person who is 65 inches (165.1 centimeters) tall and weighs 154 pounds (70 kilograms).

Step-by-Step Solution

Answer

The surface area is approximately 1.857 square meters.

Step 1: Identify Variables

First, recognize that we need to determine the variables for the formula: weight \(w\) in kilograms and height \(h\) in centimeters. In this problem, \(w = 70\) kg and \(h = 165.1\) cm.

Step 2: Set Up the Formula

Use the given formula, \(S(w, h) = 0.007184 w^{0.425} h^{0.725}\), where \(w\) is the weight in kilograms and \(h\) is the height in centimeters.

Step 3: Substitute the Variables

Substitute the given values of weight and height into the formula: \(S(70, 165.1) = 0.007184 \times 70^{0.425} \times 165.1^{0.725}\).

Step 4: Calculate Each Component

Calculate \(70^{0.425}\) and \(165.1^{0.725}\) separately. First, \(70^{0.425} \approx 7.2202\) and then \(165.1^{0.725} \approx 49.6515\).

Step 5: Apply the Multiplier and Simplify

Multiply the results from Step 4 by 0.007184: \(0.007184 \times 7.2202 \times 49.6515 \approx 1.8572\).

Step 6: Conclusion

The estimated surface area of the person is approximately 1.857 square meters.

Key Concepts

Human Body Measurements

Understanding human body measurements is crucial for calculating various metrics like body surface area. These measurements typically include weight and height, which are critical in many medical and physiological calculations. In this particular formula for surface area, weight is measured in kilograms and height in centimeters. It's vital to ensure the measurements for height and weight are accurate, as small discrepancies can lead to significant differences in the result.

When discussing these measurements, it's important to be aware that different contexts may require different units. In some countries, height might be commonly measured in inches or feet and weight in pounds. Having a clear understanding of how these conversions work is key to applying the formula correctly.

Mathematical Formulas

Mathematical formulas are essential tools for solving complex problems efficiently. The formula used in our exercise to calculate the surface area of the human body is: \(S(w, h) = 0.007184 \, w^{0.425} \, h^{0.725}\), where \(w\) is weight in kilograms and \(h\) is height in centimeters.

This specific formula originates from a scientifically-derived estimation for calculating body surface area, which is a function determined by variables that include more than just simple multiplication. These exponents, 0.425 for weight and 0.725 for height, reflect the empirical relationships between these measurements and the actual surface area.

To use mathematical formulas effectively, it's essential to understand how to substitute the right values and accurately perform exponential calculations. It is also crucial to know the role of each component in the equation to ensure a solid grasp of how the whole formula delivers its results.

Unit Conversion

Unit conversion is an essential step in problem-solving when measurements are given in different units than those required by a formula. Converting units involves using conversion factors to translate from one unit system to another.

In this example, we needed to express the person's height in centimeters and weight in kilograms to use in the formula. This required converting 65 inches to 165.1 centimeters and 154 pounds to 70 kilograms. Knowing the right conversion factors is crucial. For instance, 1 inch equals 2.54 centimeters, and 1 pound equals approximately 0.453592 kilograms.

Accurate unit conversions ensure that the mathematical calculations are correct, ultimately leading to accurate results. It's a good practice to double-check your conversions, as errors in this step can lead to incorrect outcomes, affecting the overall conclusion.