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Chapter 15: Problem 70

The body mass index (BMI) for an adult human is given by the function \(B=\frac{w}{h^{2}},\) where \(w\) is the weight measured in kilograms and \(h\) is the height measured in meters. a. Find the rate of change of the BMI with respect to weight at a constant height. b. For fixed \(h\), is the BMI an increasing or decreasing function of w? Explain. c. Find the rate of change of the BMI with respect to height at a constant weight. d. For fixed \(w,\) is the BMI an increasing or decreasing function of \(h ?\) Explain.

Short Answer

Expert verified
Answer: The BMI is an increasing function of weight and a decreasing function of height. As weight increases, BMI increases. As height increases, BMI decreases.

Step by step solution

01

Find \(\frac{\partial B}{\partial w}\)

To find the rate of change of the BMI with respect to weight, we need to find the partial derivative of B with respect to w while keeping h constant. This can be done using the following formula: $$\frac{\partial B}{\partial w} = \frac{\partial}{\partial w}\left(\frac{w}{h^{2}}\right)$$ **Step 2: Compute the partial derivative with respect to w**
02

Compute \(\frac{\partial B}{\partial w}\)

Using the quotient rule, we get: $$\frac{\partial B}{\partial w} = \frac{h^{2}}{h^{2}} = 1$$ (a) The rate of change of the BMI with respect to weight at a constant height is 1. **Step 3: Analyze the function for fixed h for increasing or decreasing**
03

BMI as a function of w

Since \(\frac{\partial B}{\partial w} = 1 > 0\), the BMI is an increasing function of weight w for fixed height h. This means that as the weight increases, the BMI also increases. **Step 4: Find the partial derivative of B with respect to h**
04

Find \(\frac{\partial B}{\partial h}\)

To find the rate of change of the BMI with respect to height, we need to find the partial derivative of B with respect to h while keeping the w constant. This can be done using the following formula: $$\frac{\partial B}{\partial h} = \frac{\partial}{\partial h}\left(\frac{w}{h^{2}}\right)$$ **Step 5: Compute the partial derivative with respect to h**
05

Compute \(\frac{\partial B}{\partial h}\)

Using the quotient rule, we get: $$\frac{\partial B}{\partial h} = \frac{-2w}{h^{3}}$$ (c) The rate of change of the BMI with respect to height at a constant weight is \(\frac{-2w}{h^{3}}\). **Step 6: Analyze the function for fixed w for increasing or decreasing**
06

BMI as a function of h

Since \(\frac{\partial B}{\partial h} = \frac{-2w}{h^{3}} < 0\), the BMI is a decreasing function of height h for fixed weight w. This means that as the height increases, the BMI decreases.

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Key Concepts

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

Body Mass Index
Body Mass Index (BMI) is a simple calculation utilizing a person's weight and height to determine their body fat level. The formula is given by:
\[ B = \frac{w}{h^2} \]where \( B \) is the BMI, \( w \) is the weight in kilograms, and \( h \) is the height in meters.
  • This index provides insight into whether an individual may have a healthy weight relative to their height.
  • It is a quick and useful measure for assessing health risk levels related to body weight.
Despite its usefulness, BMI does not directly assess body fat and can sometimes misclassify muscular individuals as overweight.
Rate of Change
The rate of change refers to how a function changes as its input changes. In the context of BMI, we often want to know how BMI changes with respect to changes in either weight \( w \) or height \( h \). This is where partial derivatives come into play.
  • Partial derivatives allow us to focus on the change of a multivariable function concerning one independent variable at a time.
  • For example, the partial derivative of BMI with respect to weight at constant height is found by keeping \( h \) constant and differentiating the function \( B \).
Finding these rates of change helps us understand how sensitive the BMI is to changes in weight and height.
Quotient Rule
The quotient rule is a technique used in calculus to find the derivative of a ratio of two functions. Since BMI is expressed as \( \frac{w}{h^2} \), the quotient rule is used to compute partial derivatives.
  • When finding \( \frac{\partial B}{\partial w} \):
    • The function in the numerator (\( w \)) is differentiated while treating the denominator (\( h^2 \)) as a constant.
  • When finding \( \frac{\partial B}{\partial h} \):
    • The formula \( f'(x)g(x) - f(x)g'(x) \) is adapted, treating \( w \) as constant and differentiating\(- \frac{w}{h^2} \) relatively to \( h \).
Mastering the quotient rule is essential for tackling functions like BMI that involve division in their structure.
Increasing and Decreasing Functions
Understanding whether a function is increasing or decreasing is essential for analyzing its behavior as inputs change. For the BMI function:
  • When \( \frac{\partial B}{\partial w} = 1 \), BMI is an increasing function of \( w \). This indicates that an increase in weight results in a proportional increase in BMI for a constant height.
  • Conversely, when \( \frac{\partial B}{\partial h} = \frac{-2w}{h^3} \), it indicates BMI is a decreasing function with respect to \( h \). Hence, as height increases, BMI decreases, assuming a constant weight.
Recognizing these trends helps individuals and healthcare professionals anticipate BMI changes based on modifications in height or weight.

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