Problem 23 - Algebra and Trigonometry with Analytic Geometry

Question

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) \( W = kh^3 \). (b) \( k \approx 0.000536 \). (c) 154 pounds.

Step 1: Understand Direct Proportionality

Since the threshold weight \( W \) is directly proportional to the cube of the height \( h \), we can write the relationship as \( W = kh^3 \), where \( k \) is the constant of proportionality. This shows how the weight varies with height, assuming \( h \) is measured in consistent units.

Step 2: Convert the Given Height to a Standard Unit

To find the constant \( k \), we need to first convert all height measurements to a consistent unit, such as inches. Since 1 foot is 12 inches, a 6-foot height is equivalent to \( 6 \times 12 = 72 \) inches.

Step 3: Find the Constant of Proportionality \( k \)

Given that \( W = 200 \) pounds for a 6-foot individual (72 inches), substitute \( W = 200 \) and \( h = 72 \) into the equation \( W = kh^3 \) to find \( k \): \[200 = k(72)^3\]Calculate \( (72)^3 = 373248 \). Therefore, \[k = \frac{200}{373248} \approx 0.000536\]

Step 4: Estimate the Threshold Weight for a Different Height

Convert the height of 5 feet 6 inches to inches: \( 5 \times 12 + 6 = 66 \) inches. Now, use the equation \( W = kh^3 \) to find \( W \) for this height:\[W = 0.000536 \times (66)^3\]Calculate \( (66)^3 = 287496 \). Substitute this into the equation:\[W = 0.000536 \times 287496 \approx 154\]Therefore, the threshold weight is approximately 154 pounds.

Key Concepts

constant of proportionality

In mathematics, the constant of proportionality is a critical concept when dealing with direct proportionality. It acts as a bridge between the two quantities that are directly connected.
To understand this, consider the expression of the threshold weight in the exercise: \( W = kh^3 \), where \( W \) represents the threshold weight, \( h \) is the height, and \( k \) is the constant of proportionality.
This constant \( k \) defines how the threshold weight scales with the height cubed. Its value determines the rate at which \( W \) changes in response to \( h \).
This means:

• The relationship between \( W \) and \( h \) will be scaled by \( k \).
• A larger \( k \) value means a greater increase in \( W \) for an increase in \( h \).

To find \( k \), we use a known situation such as a height of 6 feet (or 72 inches) and a threshold weight of 200 pounds. By substituting these values into the equation and solving for \( k \), we find that \( k \approx 0.000536 \). This process helps us apply the relationship to different cases, ensuring that the proportion holds true for various heights.

threshold weight

Threshold weight is a term used in health assessments, particularly concerning the risk associated with different body weights. It refers to the weight beyond which the risk of health-related complications, including mortality, increases significantly. This concept is explored in the exercise where the threshold weight \( W \) is specifically tied to a person's height, which is a common approach in health studies.
For example, in the exercise, the threshold weight for a 6-foot male is established at 200 pounds. Understanding the threshold weight helps in identifying the range of healthy weights for different individuals based on their unique characteristics, such as height.

• It's important to keep the units consistent. Here, height is in inches while weight is in pounds.
• The exploration of threshold weight connects biometric factors to potential health risks.

Through the understanding of threshold weight, one can determine healthier weight ranges and prevent potential health risks by recognizing when a certain weight can be riskier based on the height of an individual.

cubic relationship in math

A cubic relationship in math involves one variable being related to the cube of another. This means that the first variable scales with the cube of the second variable. In simpler terms, when one of them changes, it does so in a manner proportional to the cube of the other.
Such a relationship is evident in the formula \( W = kh^3 \) from our exercise. Here, the threshold weight \( W \) changes with the cube of the height \( h \). The cubic power signifies that a small change in height can result in a significant change in the weight, especially since cubing a number amplifies its size.

• In a cubic relationship, if height doubles, weight increases by eight times \((2^3)\).
• This can be visualized as a steep increase, common in natural phenomena like volumes.

Understanding this cubic dependence helps in appreciating how body dimensions scale differently from linear relationships, providing insight into more complex real-world interactions.