Problem 53 - Algebra 2

Question

HEALTH The prediction equation \(y=205-0.5 x\) relates a person's maximum heart rate for exercise \(y\) and age \(x\) . Use the equation to find the maximum heart rate for an 18 -year-old.

Step-by-Step Solution

Answer

The maximum heart rate for an 18-year-old is 196 bpm.

Step 1: Understanding the Equation

The prediction equation given is \( y = 205 - 0.5x \). In this equation, \( y \) represents the maximum heart rate and \( x \) is the age of the person.

Step 2: Substitute Age into the Equation

We are asked to find the maximum heart rate for an 18-year-old. Thus, substitute \( x = 18 \) into the equation: \( y = 205 - 0.5(18) \).

Step 3: Perform the Multiplication

Calculate \( 0.5 \times 18 \), which results in \( 9 \).

Step 4: Subtraction to Find Maximum Heart Rate

Subtract the result of the multiplication from 205. Thus, \( 205 - 9 = 196 \).

Step 5: Interpret the Result

The value obtained, 196, represents the maximum heart rate for exercise for an 18-year-old.

Key Concepts

Maximum Heart Rate Prediction

The maximum heart rate prediction is an important concept in the field of exercise science. It is used to estimate the maximum number of heart beats per minute a person can achieve safely during physical exertion. The commonly used formula for this prediction is \( y = 205 - 0.5x \), where \( y \) is the maximum heart rate, and \( x \) is the individual's age. This linear equation suggests that as a person ages, their maximum heart rate tends to decrease. Knowing one's maximum heart rate helps in designing safe and effective exercise programs.

• For younger individuals, the predicted maximum heart rate is higher, which allows them to exercise at a more intense level.
• As people grow older, the decrease in maximum heart rate means they should be more cautious to avoid overexertion.

This straightforward equation provides an easy way to tailor fitness routines based on age, ensuring that individuals can achieve their fitness goals while minimizing the risk of injury.

Substitution Method

The substitution method is a fundamental algebraic technique used to solve equations by replacing a variable with a given value. In this particular exercise, we use the substitution method to find a solution for the maximum heart rate prediction equation.

To apply the substitution method, follow these simple steps:

• Identify the variable you need to substitute: In this case, age \( x \) is substituted into the equation.
• Substitute the provided value into the equation: For an 18-year-old, we substitute \( x = 18 \) into the equation \( y = 205 - 0.5x \).
• Simplify the resulting equation: After substitution, perform the necessary arithmetic operations on the equation.

This approach is helpful for making calculations easier and is widely applied in solving various types of linear equations.

Mathematical Modeling

Mathematical modeling involves using equations to represent real-world situations and predict outcomes. In this exercise, the equation \( y = 205 - 0.5x \) serves as a mathematical model for estimating maximum heart rate based on a person's age.

Mathematical models like this one offer several benefits:

• They provide a simplified representation of complex phenomena, making them easier to understand and utilize.
• Models help in making informed decisions, such as determining safe exercise levels based on age.
• They are flexible and can be adjusted as new data becomes available, allowing for more accurate predictions.

Using mathematical models in health and fitness allows trainers and individuals to develop personalized exercise plans that effectively balance intensity and safety.