Problem 63 - College Algebra and Calculus: An Applied Approach

Question

Health A person's maximum heart rate is \(220-x\), where \(x\) is the person's age in years for \(20 \leq x \leq 70\). When a person exercises, it is recommended that the person strive for a heart rate that is at least \(50 \%\) of the maximum and at most \(75 \%\) of the maximum. (Source: American Heart Association) (a) Write a system of inequalities that describes the exercise target heart rate region. Let \(y\) represent a person's heart rate. (b) Sketch a graph of the region in part (a). (c) Find two solutions to the system and interpret their meanings in the context of the problem.

Step-by-Step Solution

Answer

The system of inequalities that describes the target heart rate for exercise is \(0.50(220 - x) \leq y \leq 0.75(220 - x)\) for \(20 \leq x \leq 70\). The graph illustrates that the target heart rate decreases as a person gets older. Two solutions are for ages 30 and 50 years, the target heart rates range between 95 - 142.5 and 85 - 127.5 respectively.

Step 1: Write the system of inequalities

The maximum heart rate is given by the expression \(220 - x\). The minimum and maximum targets for the exercise heart rate are \(50 \%\) and \(75 \%\) of the maximum respectively. These can be written as \(0.50(220 - x)\) and \(0.75(220 - x)\). Therefore, the system of inequalities is: \(0.50(220 - x) \leq y \leq 0.75(220 - x)\) and \(20 \leq x \leq 70\).

Step 2: Graph the inequalities

To sketch the inequalities, you should plot two lines representing the equations \(y = 0.50(220 - x)\) and \(y = 0.75(220 - x)\). The region between them is the target heart rate region. The interval for \(x\) is from 20 to 70, thus, the region of interest is between those values.

Step 3: Find two solutions

The solutions to this system are the pairs \((x, y)\) that satisfy the inequalities. For instance, for \(x = 30\) years, the target heart rate range is between \(0.50(220 - 30) = 95\) and \(0.75(220 - 30) = 142.5\). For \(x = 50\) years, the target heart rate is between \(0.50(220 - 50) = 85\) and \(0.75(220 - 50) = 127.5\). Thus, two solutions are \((30, 95)\), \((30, 142.5)\), \((50, 85)\) and \((50, 127.5)\).

Key Concepts

Maximum Heart Rate

To understand maximum heart rate, it's important to know that it estimates the highest number of times your heart can safely beat per minute during physical activity. This is expressed as \(220 - x\), where \(x\) represents your age in years. Thus, as you get older, your maximum heart rate gradually decreases, providing a safer gauge for exercise.

• For example, a 30-year-old has a maximum heart rate of \(220 - 30 = 190\) beats per minute.
• A 50-year-old would have a reduced rate of \(220 - 50 = 170\) beats per minute.

This formula helps in setting personal exercise goals and understanding the limits during physical activities. Staying within these limits ensures cardiovascular effectiveness without overexerting the heart.

Exercise Heart Rate

When you exercise, hitting the right heart rate is important for effective and safe workouts. Targeting an exercise heart rate means achieving a balance between intensity and safety. The exercise heart rate lies between 50% and 75% of your maximum heart rate. To calculate:

• Minimum exercise heart rate: \(0.50(220 - x)\)
• Maximum exercise heart rate: \(0.75(220 - x)\)

For a 30-year-old, this means keeping the heart rate betweenul>

\(95\) (50% of \(190\))

\(142.5\) (75% of \(190\))

By staying within these values, the person receives the full benefits of the workout while reducing the risk of heart-related incidents.

System of Inequalities

Solving a system of inequalities involves determining a range of values that fit certain conditions. In this case, we have two main elements:- The age range of \(20 \leq x \leq 70\).- The heart rate range of \(0.50(220 - x) \leq y \leq 0.75(220 - x)\).To visualize this, one would graph two lines:

• \(y = 0.50(220 - x)\)
• \(y = 0.75(220 - x)\)

Then, identify the region between these lines, which represents the exercise heart rate zone for the given age limits.For any \(x\) value within the age range, the corresponding \(y\) value should fit between the two calculated values for heart rate. This visual representation helps in identifying acceptable exercise heart rates throughout the age spectrum from 20 to 70.