Problem 47 - 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 6 feet tall weighing 205 pounds within the healthy weight region?

Step-by-Step Solution

Answer

No, a person in this age group who is 6 feet tall and weighs 205 pounds is not within the healthy weight region.

Step 1: Convert height from feet to inches

Since 1 foot is equal to 12 inches, therefore 6 feet would be equal to \(6*12 = 72 inches\). So, x for the given person is 72.

Step 2: Apply the given x in the inequalities

Substitute x = 72 in the given linear inequalities. Eq1: \(5.3*72 - y \geq 180\); Eq2: \(4.1*72 - y \leq 140\)

Step 3: Solve these inequalities for y

Solving these inequalities will provide us with a weight range for the given person's height. Eq1 becomes \(381.6 - y \geq 180\) which simplifies to \(y \leq 201.6\) ; Eq2 simplifies to \(295.2 - y \leq 140\), which simplifies to \(y \geq 155.2\)

Step 4: Compare the obtained weight range to the person's weight

The given person's weight y = 205 pounds. The solved inequalities tells us that the weight should be less than or equal to 201.6 pounds and more than or equal to 155.2 pounds for the person to be in the healthy weight region. But, the given weight 205 is greater than 201.6, hence the person lies outside the healthy weight region.

Key Concepts

Healthy Weight Region

The concept of a healthy weight region helps us understand how weight is related to height in evaluating health. This evaluation uses mathematical models to set boundaries within which a person's weight is considered healthy according to their height. For example, using inequalities, we can define this region as the set of possible weight values for a given height.

In the exercise, we're asked to use the system of inequalities to identify if a particular height and weight pair falls within this healthy region. When someone measures 72 inches tall (or 6 feet) and weighs 205 pounds, we apply these boundaries to determine if this weight heightens health concerns.

• We calculate if the weight satisfies both inequalities.
• If it does, the person is inside the healthy weight region.
• If not, they lie outside of it.

Ultimately, understanding the healthy weight region encourages better health management.

System of Inequalities

A system of inequalities consists of multiple inequality expressions that share the same variables. They help define regions or ranges of values that satisfy all given conditions simultaneously. In our context, they are used to show combinations of height and weight that are considered healthy.

For the given problem, the system is composed of two inequalities:

• \(5.3x - y \geq 180\)
• \(4.1x - y \leq 140\)

These inequalities provide the upper and lower bounds for weight in relation to height. Solving them requires testing the conditions to see if a specific pair of values lies within the indicated area. When both inequalities are satisfied, the height and weight combination is part of the defined healthy region.

Linear Equations

Linear equations represent relationships between variables involving straight-line graphs. In this exercise, linear equations, in the form of inequalities, describe the healthy weight boundaries for different heights.

Linear equations are straightforward and all solutions form a line when graphed. In the inequalities:

• \(5.3x - y = 180\)
• \(4.1x - y = 140\)

These form the basis of the healthy weight region, with linear inequalities shaping a specific area on the graph which reflects these boundaries.Understanding linear equations, even as inequalities, empowers learners to predict and evaluate outcomes based on variable interactions.

Mathematical Modeling

Mathematical modeling uses mathematical descriptions to represent real-life scenarios, like understanding whether certain height and weight points fall into a healthy category.

The exercise here models the healthy weight region as a system of linear inequalities. This method allows us to simulate health guidelines in a precise, mathematical form, bridging the gap between numbers and health standards.

• It aids in estimating health conditions based on calculable standards.
• Simulates scenarios and solutions for given parameters.
• Offers visual and numerical insights into how factors, like weight, are influenced by height.

By learning mathematical modeling, we can apply these principles to a multitude of practical applications, embracing a systematic approach to problem-solving.