Problem 48 - Thinking Mathematically

Question

If \(x\) represents height, in inches, and y represents weight, in pounds, the healthy weight region can be modeled by the following system of linear inequalities: $$ \left\\{\begin{array}{l} 5.3 x-y \geq 180 \\ 4.1 x-y \leq 140 \end{array}\right. $$ Use this information to solve Exercises 45-48. Is a person in this age group who is 5 feet 8 inches tall weighing 135 pounds within the healthy weight region?

Step-by-Step Solution

Answer

The short answer will be a statement that indicates whether or not the person is within the healthy weight region according to the outcomes of the evaluated inequalities.

Step 1: Convert feet to inches

The first step is to convert height from feet to inches. This is done because the 'x' variable in our inequalities represents height in inches. Therefore, using this conversion: 1 foot = 12 inches, a height of 5 feet 8 inches corresponds to \(5*12 + 8 = 68\) inches.

Step 2: Substitute height and weight

Then, substitute the height (68 inches) into 'x' and weight (135 pounds) into 'y' in both inequalities and check if both conditions are satisfied: \[5.3*68 - 135 \geq 180\] and \[4.1*68 - 135 \leq 140\].

Step 3: Evaluate the inequalities

Now, evaluate both inequalities to identify whether the person falls into the healthy weight region. If both inequalities are true, the weight of the person at the given height is considered healthy.

Key Concepts

Understanding System of Inequalities

A system of inequalities involves solving multiple inequalities simultaneously. In this exercise, we have a system with two inequalities that model a healthy weight region based on height and weight. Each inequality represents a boundary in a two-dimensional space.

When solving these, the intersection of the solution sets gives the region where both conditions are satisfied.

• The first inequality, \(5.3x - y \geq 180\), suggests a minimum weight boundary.
• The second inequality, \(4.1x - y \leq 140\), sets a maximum weight boundary.

For a set of values \((x,y)\) to satisfy the system, they must fall within the region defined by these inequalities. Solving a system means determining where both inequalities are true, providing a range of acceptable solutions or conditions. This concept is essential in various fields, including health, to ensure certain standards are maintained.

Height and Weight Conversion: From Feet to Inches

Before we can substitute values into our inequalities, it is crucial to convert measurements into the correct units. In American customary units, height is often measured in feet and inches, which can be confusing when dealing with calculations. Here, our variable \(x\) represents height in inches. Therefore, conversion is necessary.

To convert a height given in feet and inches to inches alone:

• Multiply the number of feet by 12 (since 1 foot is equal to 12 inches).
• Add the remaining number of inches to this product.

For example, converting 5 feet 8 inches:

• Calculate \(5 \times 12 + 8 = 68\) inches.

This conversion simplifies inputs into our system of inequalities, ensuring units are consistent. Correct unit conversion is a critical step in ensuring that mathematical models and real-world measurements align accurately.

Evaluating the Inequalities: Testing Conditions

Once you have substituted the appropriate values into the inequalities, the next task is to check each one. Evaluating inequalities means determining whether the conditions specified by each inequality hold true for a given set of values, such as height and weight.

To evaluate if a specific person falls within the defined healthy weight region, substitute the converted height and weight into each inequality:

• For \(5.3 \times 68 - 135 \geq 180\), perform the calculation to see if the inequality holds true.
• For \(4.1 \times 68 - 135 \leq 140\), check if this condition is also met.

If both conditions are satisfied, the person falls within the intended region. This two-step evaluation helps to comprehensively determine whether the individual is in a desired or acceptable state, based on defined mathematical parameters.

Mathematical Modeling: Real-World Applications

Mathematical modeling is a way of representing real-world situations through mathematical expressions or equations. In this exercise, the system of inequalities serves as a model for determining a healthy weight region, based on height and weight.

Models like these are valuable in providing a simplified representation of complex phenomena, such as:

• Health and nutrition guidelines.
• Engineering systems.
• Environmental estimations.

In particular, the use of inequalities allows for ranges rather than absolute figures, reflecting the variability in safe or healthy standards. Mathematical models, when well constructed, can predict, inform, and guide decisions in personal and professional domains. They offer a structured way to handle uncertainty and make predictions based on known data and understood relationships.