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Chapter 3: Problem 24

A typical male's body surface area \(S\) in square meters is often modeled by the formula \(S=\frac{1}{60} \sqrt{w h},\) where \(h\) is the height in centimeters, and \(w\) the weight in kilograms, of the person. Find the rate of change of body surface area with respect to weight for males of constant height \(\left.h=180 \mathrm{cm} \text { (roughly } 5^{\prime} 9^{\prime \prime}\right)\) Does \(S\) increase more rapidly with respect to weight at lower or Higher body weights? Explain.

Short Answer

Expert verified
The rate of change decreases as weight increases; \( S \) increases more rapidly at lower weights.

Step by step solution

01

Understand the Formula

The body surface area formula is given as \( S = \frac{1}{60} \sqrt{w h} \). We need to find the rate of change of \( S \) with respect to weight \( w \) while keeping the height \( h \) constant at 180 cm.
02

Substitute Constant Height

Substitute the constant height value \( h = 180 \) cm into the formula: \( S = \frac{1}{60} \sqrt{w \times 180} = \frac{1}{60} \sqrt{180w} \).
03

Differentiate with Respect to Weight

Find the derivative of \( S \) with respect to \( w \). Start with \( S = \frac{1}{60} \, (180w)^{1/2} \). Differentiating gives: \( \frac{dS}{dw} = \frac{1}{60} \times \frac{1}{2} \times (180w)^{-1/2} \times 180 \).
04

Simplify the Derivative

Simplify the derivative: \( \frac{dS}{dw} = \frac{1}{60} \times \frac{180}{2} \times \frac{1}{\sqrt{180w}} = \frac{3}{2 \sqrt{180w}} \).
05

Analyze Rate of Change

Since \( \frac{dS}{dw} = \frac{3}{2 \sqrt{180w}} \), as \( w \) increases, the denominator (\( \sqrt{180w} \)) becomes larger, making \( \frac{dS}{dw} \) smaller. Therefore, the rate of change of \( S \) with respect to \( w \) decreases as \( w \) increases.

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Key Concepts

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

Rate of Change
In calculus, the rate of change is a fundamental concept that describes how a quantity changes with respect to another. In this exercise, we aim to find how the body surface area changes as a person's weight varies, while keeping their height constant. This is described mathematically by the derivative. Understanding how the body surface area reacts to changes in weight is crucial for many applications, such as medicine and physiology. It helps us see how quickly or slowly the body surface area increases when weight is gained. It answers the practical question: "Does a lighter person experience a greater change in their body area for a small weight gain compared to someone heavier?" The rate of change, in this instance, is a measure that gives us insight into whether adding more weight will have a significant impact on the body surface area. Essentially, it tells us that for light individuals, body surface area increases rapidly with weight gain, while for heavier individuals, the increase is more gradual.
Body Surface Area
Body surface area (BSA) is an important measurement that reflects the external surface of a human body. It's widely used in various medical applications, including prescribing correct drug dosages and assessing nutritional needs.The calculation of BSA can be complex, and different formulas exist. In this exercise, the Mosteller formula is used: \( S = \frac{1}{60} \sqrt{wh} \). This formula takes into account both height and weight to provide a square meter measurement for body surface area.Using BSA instead of weight and height alone can provide a more accurate representation of physiological processes. For instance, in medical practices, it helps in better drug dosing because it correlates more closely with metabolic mass. When considering changes in BSA, it's essential to understand its sensitivity to changes in the person's weight and height, as displayed in this exercise.
Derivative Calculation
Derivative calculation is key in understanding how functions change. It gives us the rate of change or the slope of a curve at any given point. Calculating the derivative involves applying differentiation rules to find how a dependent variable (in this case, body surface area \( S \)) changes with respect to an independent variable (weight \( w \)).To calculate the derivative of body surface area with respect to weight, we start by substituting the constant height into our formula, simplifying it so that we differentiate only with respect to \( w \). The derivative, \( \frac{dS}{dw} \), is calculated as \( \frac{3}{2 \sqrt{180w}} \). This represents the rate at which body surface area changes per unit change in weight.The derivative shows that as weight \( w \) increases, the change in body surface area grows more slowly. This indicates a diminishing rate of change, a critical insight for understanding how weight variations impact an individual鈥檚 body surface.

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Most popular questions from this chapter

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