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Chapter 12: Problem 54

The formula $$S=0.007184 W^{0.425} H^{0.725}$$ gives the surface area \(S\) of a human body (in square meters) in terms of its weight \(W\) (in kilograms) and its height \(H\) (in centimeters). Compute \(\partial S / \partial W\) and \(\partial S / \partial H\) when \(W=70 \mathrm{~kg}\) and \(H=180 \mathrm{~cm}\). Interpret your results.

Short Answer

Expert verified
We find that \(\frac{\partial S}{\partial W} = 0.007184 \times 0.425 (70)^{-0.575} (180)^{0.725} \approx 0.00887\), and \(\frac{\partial S}{\partial H} = 0.007184 \times 0.725 (70)^{0.425} (180)^{-0.275} \approx 0.01882.\) Both partial derivatives are positive, which means that the surface area increases as weight and height increase. Specifically, when the weight increases by 1 kg, the surface area increases by about 0.00887 square meters, and when the height increases by 1 cm, the surface area increases by about 0.01882 square meters.

Step by step solution

01

Compute \(\frac{\partial S}{\partial W}\)

To compute the partial derivative of S with respect to W, we keep the variable H constant and differentiate with respect to W: \(\frac{\partial S}{\partial W} = 0.007184 \times 0.425 W^{0.425 - 1} H^{0.725}\) Now, substitute the given values of W and H: \(\frac{\partial S}{\partial W} = 0.007184 \times 0.425 (70)^{-0.575} (180)^{0.725}\) Evaluate this expression to get the final value of \(\frac{\partial S}{\partial W}\).
02

Compute \(\frac{\partial S}{\partial H}\)

To compute the partial derivative of S with respect to H, we keep the variable W constant and differentiate with respect to H: \(\frac{\partial S}{\partial H} = 0.007184 \times 0.725 W^{0.425} H^{0.725-1}\) Now, substitute the given values of W and H: \(\frac{\partial S}{\partial H} = 0.007184 \times 0.725 (70)^{0.425} (180)^{-0.275}\) Evaluate this expression to get the final value of \(\frac{\partial S}{\partial H}\).
03

Interpret the results

The values of \(\frac{\partial S}{\partial W}\) and \(\frac{\partial S}{\partial H}\) tell us how the surface area of the human body changes with respect to changes in weight and height. A positive value for \(\frac{\partial S}{\partial W}\) indicates that the surface area increases as the weight increases. A negative value for \(\frac{\partial S}{\partial W}\) indicates that the surface area decreases as the weight increases. A positive value for \(\frac{\partial S}{\partial H}\) indicates that the surface area increases as the height increases. A negative value for \(\frac{\partial S}{\partial H}\) indicates that the surface area decreases as the height increases. As you may have already guessed, based on the biological intuition, increasing weight and height should both result in positive values for their respective partial derivatives, suggesting that the surface area increases in both cases.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Applied Mathematics in Understanding Human Body Surface Area
Applied mathematics is all about employing mathematical methods and principles to solve real-world problems, and a perfect example of this is the calculation of human body surface area. In the problem given, we utilize a mathematical formula, which is a function of an individual's weight and height, to ascertain the surface area.

Partial derivatives come into play in applied mathematics when we want to understand how a function changes as one of its variables changes while keeping the other variables constant. Utilizing partial derivatives of the body surface area formula with respect to weight and height provides us with insight into how slight variations in weight or height can affect the surface area of a human body. This understanding is critical in fields such as medicine, where dosages of drugs might be directly proportional to a patient's body surface area.

By computing \(\frac{\partial S}{\partial W}\) and \(\frac{\partial S}{\partial H}\), we are analyzing the sensitivity of surface area to changes in weight and height, respectively. These calculations are representative of the practical utility of applied mathematics and highlight the importance of mathematical modeling in biological and medical contexts.
Understanding Human Body Surface Area and Its Implications
Human body surface area (BSA) is a measure widely used in several medical practices, including calculating drug dosages and medical indicators like body mass index (BMI). The body surface area is thought to be a better indicator of metabolic mass than body weight because it's less affected by abnormal adipose mass. Hence, understanding how BSA is impacted by changes in weight (\(W\)) and height (\(H\)) can have significant applications.

The formula used in the exercise, \(S=0.007184 W^{0.425} H^{0.725}\), is known as the Dubois formula, one of the most classic estimations of BSA. Insights gained from computing partial derivatives, such as \(\frac{\partial S}{\partial W}\) and \(\frac{\partial S}{\partial H}\), are beneficial for adjusting treatments to the unique physiologies of patients. A higher \(\frac{\partial S}{\partial W}\) would indicate that a minor increase in weight sharply increases the surface area, which may entail a significant change in dosage for medications that are dosed based on BSA.
Mathematical Modeling: A Tool for Prediction and Analysis
Mathematical modeling is an invaluable tool in predicting and analyzing complex systems, providing a framework to understand variables and their relationships. In medical science, modeling equations like the one for BSA allow us to explore the human body in a quantifiable way. Mathematical models are constructed to represent systems鈥攆rom the spread of diseases to the metabolic functions of the body鈥攊n a simplified form that is more amenable to mathematical analysis and insight.

In our specific case, the model for calculating BSA simplifies the intricacies of human physiology into an equation involving weight and height. This simplification enables health professionals to make informed decisions based on relatively easy measurements. The partial derivatives calculated as part of this model tell us how small changes in the inputs鈥攚eight and height鈥攃an influence the output, i.e., BSA, a critical piece of information when adjusting healthcare strategies. Thus, mathematical modeling not only assists in understanding current conditions but also equips professionals with predictive capabilities that are crucial in planning effective interventions.

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Most popular questions from this chapter

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