Problem 23 - Precalculus: Functions and Graphs

Question

Threshold weight Threshold weight \(W\) is defined to be that weight beyond which risk of death increases significantly. For middle-aged males, \(W\) is directly proportional to the third power of the height \(h\) (a) Express \(W\) as a function of \(h\) by means of a formula that involves a constant of proportionality \(k\) (b) For a 6 -foot male, \(W\) is about 200 pounds. Find the value of \(k\) in part (a). (c) Estimate, to the nearest pound, the threshold weight for an individual who is 5 feet 6 inches tall.

Step-by-Step Solution

Answer

(a) \(W = k \, h^3\) (b) \(k \approx 0.000536\) (c) 154 pounds.

Step 1: Understanding the Relationship

Given that the threshold weight, \(W\), is directly proportional to the third power of the height, \(h\), we can express this relationship using the formula \(W = k \, h^3\), where \(k\) is the constant of proportionality.

Step 2: Finding the Constant of Proportionality

We are told that for a 6-foot male, \(W = 200\) pounds. First, we need to convert the height into inches since the standard unit for such calculations is often in inches. 6 feet equals 72 inches. We substitute into the formula: \[ 200 = k \, (72)^3 \]Solving for \(k\), we get: \[ k = \frac{200}{72^3} \]

Step 3: Calculate the Constant k

Compute \(72^3\):\[ 72^3 = 373248 \]Now calculate \(k\):\[ k = \frac{200}{373248} \approx 0.000536 \]

Step 4: Estimate Threshold Weight for 5 Feet 6 Inches

Convert 5 feet 6 inches to inches. 5 feet is 60 inches and adding 6 inches gives 66 inches. Use the formula:\[ W = 0.000536 \, (66)^3 \]

Step 5: Calculate the Threshold Weight

Compute \(66^3\):\[ 66^3 = 287496 \]Then calculate \(W\):\[ W = 0.000536 \, \times \, 287496 \approx 154 \]

Step 6: Rounding to Nearest Pound

Finally, the estimated threshold weight for an individual who is 5 feet 6 inches tall is approximately 154 pounds.

Key Concepts

Constant of Proportionality

In mathematics, the constant of proportionality is a core element in defining relationships between two directly proportional quantities. In our exercise, this relationship is expressed through the formula \( W = k \cdot h^3 \). Here, \( W \) is the threshold weight, \( h \) is the height, and \( k \) is the constant of proportionality. This constant derives its value from specific conditions given in a problem.
For instance, when \( W = 200 \) pounds for a height \( h \) of 72 inches, we find \( k \) by rearranging the formula to \( k = \frac{W}{h^3} \). Calculating the value of \( k \) allows us to predict \( W \) for other heights, making it a crucial component in mathematical modeling of direct relationships.

Direct Variation

Direct variation describes a simple relationship where two variables change in tandem. This means if one variable increases, the other does as well, and the same happens when one decreases. In our example, the threshold weight \( W \) increases as a direct cubic function of the height \( h \).
Mathematically, it follows the form \( W = k \cdot h^3 \). This implies a consistent ratio between the height raised to the third power and the weight, where \( k \) maintains balance across different values of \( h \). Direct variation provides a linear approach to understanding growth patterns in contexts like physics, biology, and economy whenever such predictable relationships exist.

Mathematical Modeling

Mathematical modeling is a process of translating real-world phenomena into mathematical concepts that can be studied and analyzed. Using a model like \( W = k \cdot h^3 \), we represent a biological correlation between a person's height and their threshold weight. These models simplify complex systems into understandable structures.
They help in predicting outcomes, exploring scenarios, and making informed decisions. In our scenario, measuring threshold weight offers a pragmatic method for understanding health risks, facilitating medical assessments, and planning preventive actions.

Cubic Functions

Cubic functions are polynomial functions of degree three, described in the form \( f(x) = ax^3 + bx^2 + cx + d \). In the exercise, we observe \( W = k \cdot h^3 \) as a specialized cubic function where coefficients for terms involving lower degrees of \( h \) are zero.
This mathematical expression describes how threshold weight varies non-linearly with respect to height. Unlike linear or quadratic, cubic functions afford more flexibility, capturing intricate dependencies and variations, which can be particularly significant in modeling phenomena like volumetric growth or structural stress in engineering contexts.

Height and Weight Relationship

The relationship between height and weight is a common subject of study in various domains such as health sciences and sports. Understanding how these two factors interact is crucial for diagnosing potential risks or improvements in well-being.
In this particular model, we acknowledge height as a significant factor influencing threshold weight, encapsulated in the expression \( W = k \cdot h^3 \). By knowing this relationship, one can predict weight limits beyond which health risks increase. It serves practical applications in devising fitness regimes, nutritional plans, and medical evaluations to ensure balanced development and healthy lifestyles.