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Chapter 4: Problem 17

The body-mass index (BMI) of an individual who weighs \(w\) pounds and is \(h\) inches tall is given as $$ B=703 \frac{w}{b^{2}} $$ (Source: \(C D C\) a. Write an equation showing the relationship between the body-mass index and the height of a young teenager who weighs 100 pounds. b. Construct a related-rates equation showing the interconnection between the rates of change with respect to time of the body-mass index and the height. c. If the weight of the teenager who is 5 feet 3 inches tall remains constant at 100 pounds while she is growing at a rate of \(\frac{1}{2}\) inch per year, how quickly is her body-mass index changing?

Short Answer

Expert verified
The BMI is decreasing at a rate of approximately -0.056 per year.

Step by step solution

01

Express the Relationship Between BMI and Height

We start with the formula for body-mass index (BMI):\[ B = 703 \frac{w}{h^2} \]Given that the weight \( w = 100 \) pounds, substitute this into the equation:\[ B = 703 \frac{100}{h^2} \] This equation expresses BMI as a function of height \( h \).
02

Construct the Related-Rates Equation

To find how rates change, we need to differentiate the BMI formula with respect to time \( t \). We take the derivative of both sides:\[ \frac{d}{dt}[B] = \frac{d}{dt} \left( 703 \frac{100}{h^2} \right) \]Using the chain rule, this becomes:\[ \frac{dB}{dt} = 703 \cdot 100 \cdot \frac{-2h}{h^4} \cdot \frac{dh}{dt} \]Simplify to:\[ \frac{dB}{dt} = -\frac{140600}{h^3} \frac{dh}{dt} \]
03

Determine BMI Change Rate with Known Growth Rate

The teenager's height is \( 5 \) feet \( 3 \) inches, which we convert into inches:\[ h = 5 \times 12 + 3 = 63 \text{ inches} \]Given \( \frac{dh}{dt} = \frac{1}{2} \) inch per year, substitute \( h = 63 \) and \( \frac{dh}{dt} = \frac{1}{2} \) into the related-rates equation:\[ \frac{dB}{dt} = -\frac{140600}{63^3} \cdot \frac{1}{2} \]Compute to find:\[ \frac{dB}{dt} \approx -0.056 \] Thus, the BMI decreases at approximately \(-0.056\) per year.

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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, or BMI, is a simple way to assess whether an individual has a healthy body weight for a certain height. It's calculated by dividing a person's weight in pounds by the square of their height in inches, then multiplying by the number 703. This gives a number that can help indicate whether someone is underweight, normal weight, overweight, or obese.
  • The formula for BMI is: \[ B = 703 \frac{w}{h^2} \]where \( B \) is the BMI, \( w \) is the weight in pounds, and \( h \) is the height in inches.
  • It's important to understand that BMI is a screening tool and not a diagnostic tool. It doesn't measure body fat directly.
  • BMI can give general information about the potential health status of a population, but individual assessments should also consider other factors like muscle mass, bone density, and overall health conditions.
Differential Calculus
Differential Calculus helps us understand how a function changes as its inputs change. When dealing with a quantity that changes with respect to another, like BMI with height over time, differential calculus provides the tools to explore these changes.
  • The derivative is a key concept. It represents the rate at which one quantity changes with respect to another.
  • In our example, we use it to find out how BMI changes as height changes over time.
  • By taking the derivative of the BMI with respect to time, we're effectively using calculus to find how quickly the BMI changes with the change in height.
Rate of Change
Rate of change is all about how fast one quantity alters as another modifies. In our exercise, this idea translates into understanding how fast the BMI changes as the height of a teenager increases with time.
  • The rate of change is expressed as a derivative in calculus. In our BMI problem, it's the rate at which BMI changes per unit increase in time.
  • When we computed \[ \frac{dB}{dt} \], this depicts how fast the BMI is changing each year while the teenager grows.
  • In practical terms, if a teenager is growing taller but keeping the same weight, her BMI will decrease. This decrease is calculated using the rate of change.
Chain Rule
The chain rule is a fundamental tool in calculus used for finding the derivative of a composite function. When dealing with related rates problems, it's essential in linking two rates of change.
  • It allows us to differentiate two interconnected variables with respect to a third one, like time.
  • We applied the chain rule when we took the derivative of the BMI formula. By interpreting the BMI as a function of height and time, we assessed how small changes in height affect BMI.
  • The formula we derived, \[ \frac{dB}{dt} = -\frac{140600}{h^3} \frac{dh}{dt} \], shows how the change in height over time \( \frac{dh}{dt} \), affects the change in BMI \( \frac{dB}{dt} \).

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