Problem 93 - A Graphical Approach to College Algebra

Question

Solve each problem. The federal government has developed the body mass index (BMI) to determine ideal weights. A person's BMI is directly proportional to his or her weight in pounds and inversely proportional to the square of his or her height in inches. (A BMI of 19 to 25 corresponds to a healthy weight.) A 6-foot-tall person weighing 177 pounds has a BMI of \(24 .\) Find the BMI (to the nearest whole number) of a person whose weight is 130 pounds and whose height is 66 inches.

Step-by-Step Solution

Answer

The BMI of the person is 21.

Step 1: Understand the Formula for BMI

The Body Mass Index (BMI) can be calculated using the formula \( BMI = \frac{k \cdot W}{H^2} \), where \( W \) is the weight in pounds, \( H \) is the height in inches, and \( k \) is the constant of proportionality.

Step 2: Determine the Constant 'k'

Given a person who is 6 feet (72 inches) tall and weighs 177 pounds with a BMI of 24, find the constant \( k \) using the formula \( 24 = \frac{k \cdot 177}{72^2} \). Simplify and solve for \( k \): \( k = \frac{24 \cdot 72^2}{177} \).

Step 3: Calculate 'k' Value

Calculate \( k \) by performing the arithmetic: \( k = \frac{24 \cdot 5184}{177} \), which simplifies to \( k \approx 7.054 \).

Step 4: Apply 'k' to New Person's BMI

Using the calculated value of \( k \), find the BMI of the person who weighs 130 pounds and is 66 inches tall using the formula \( BMI = \frac{7.054 \cdot 130}{66^2} \).

Step 5: Calculate the New BMI

First, calculate \( 66^2 = 4356 \). Then, find \( BMI = \frac{7.054 \cdot 130}{4356} \), which results in \( BMI \approx 21.024 \).

Step 6: Round the BMI to the Nearest Whole Number

Round the calculated BMI of 21.024 to the nearest whole number, resulting in a BMI of 21.

Key Concepts

Proportionality in BMI

The concept of proportionality is central to understanding how the Body Mass Index (BMI) is calculated and used. When we say that the BMI is directly proportional to weight, it means that as weight increases, the BMI will increase if height remains constant. Inversely, the BMI is inversely proportional to the square of the height. This implies that, for a given weight, an increase in height will result in a decrease in BMI.

To grasp proportionality in this context, consider the formula for BMI:

• Directly proportional: BMI increases as weight (\(W\)) increases, given constant height.
• Inversely proportional: BMI decreases as height (\(H\)) increases when weight is held constant, due to the squaring of height.

These relationships are crucial for calculating and interpreting BMI accurately. Understanding how adjustments in weight or height impact BMI can help in comprehending health guidelines and assessing weight categories.

Weight and Height Relationship

The relationship between weight and height is a key element in determining one's BMI, showcasing how these two variables interact. Given that BMI is a predictive measure of body fat based on these dimensions, understanding their interaction is important for determining whether an individual is within a healthy range.

In the exercise, we observe how the height squared (\(H^2\)) plays into the calculation, emphasizing the impact of height:

• Weight (\(W\)) is in the numerator, indicating a direct impact as it increases BMI when weight goes up.
• Height squared (\(H^2\)) is in the denominator, showing that an increase in height results in a lower BMI, assuming constant weight.

This interaction suggests that people of the same weight but different heights will have different BMI values, highlighting the relative importance of height in assessing body composition. This understanding is beneficial for setting personalized health goals based on height-weight proportions.

Algebraic Problem Solving with BMI

Algebraic problem-solving skills are essential for working through BMI calculations effectively, especially when determining unknown quantities such as the constant of proportionality or a new BMI from given data. In solving these problems, algebra assists in rearranging formulas and isolating variables.

For instance, in the step-by-step solution, you are required to find the constant \(k\) using known quantities: \(24 = \frac{k \cdot 177}{72^2}\). This entails manipulating the formula to solve for \(k\):

• Multiply both sides by the height squared to isolate the term with \(k\): \(k \cdot 177 = 24 \cdot 72^2\).
• Divide both sides by the weight to solve for \(k\): \(k = \frac{24 \cdot 72^2}{177}\).
• Apply this value of \(k\) to compute BMI for different weight/height pairs.

These algebraic methods enable a clear, systematic approach to solving BMI problems, ensuring precision in calculations and accuracy in results.