Problem 59 - Algebra 2

Question

HEAIH 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.

Step 1: Identify the given equation

The prediction equation provided is \( y = 205 - 0.5x \), where \( y \) represents the maximum heart rate for exercise and \( x \) is the age of the person.

Step 2: Insert the age into the equation

We need to find the maximum heart rate for an 18-year-old. This means substituting \( x = 18 \) into the equation. So, our equation becomes \( y = 205 - 0.5(18) \).

Step 3: Simplify the multiplication

Calculate \( 0.5 \times 18 \), which equals 9. Now the equation becomes \( y = 205 - 9 \).

Step 4: Perform the subtraction

Subtract 9 from 205 to find the maximum heart rate: \( 205 - 9 = 196 \).

Step 5: State the result

The maximum heart rate for an 18-year-old is \( y = 196 \).

Key Concepts

Prediction Equation

The prediction equation is a mathematical formula used to predict or estimate an outcome based on certain variables. In our case, the equation \( y = 205 - 0.5x \) is used to estimate a person's maximum heart rate for exercise based on their age. The variable \( y \) represents the maximum heart rate, while \( x \) stands for the age of the individual. This equation is valuable in fitness and health fields to customize exercise plans. Being able to predict how a person's heart rate responds to exercise can help in setting safe and effective workout intensities.

Using such equations, trainers and healthcare professionals can gauge how hard a person should be working out, ensuring safety while maximizing workout benefits.

Age Substitution

Age substitution is a simple step in algebra where a known value is substituted into an equation. By replacing \( x \) with a specific age, we can calculate the maximum heart rate for that age. For instance, in the given problem, we substitute 18 for \( x \) to calculate the maximum heart rate for an 18-year-old, turning the equation into \( y = 205 - 0.5(18) \). This substitution simplifies the process and clarifies what needs to be solved.

Substitution is crucial because it replaces unknown variables, making it possible to perform further calculations and eventually find the desired result.

Linear Equation

A linear equation is an equation that represents a straight line when graphed. It generally has one or two variables and takes the form of \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. In the context of our prediction equation, \( y = 205 - 0.5x \), you can see it's linear because it fits this general form, with \( -0.5 \) being the slope and \( 205 \) the y-intercept.

Linear equations are easy to work with and solve because they involve simple calculations. They don't include any squared variables or complex functions, making our problem straightforward and our calculations manageable.

Algebraic Calculation

Algebraic calculations involve manipulating equations to find unknown values. Once we've substituted the age into our equation as \( y = 205 - 0.5(18) \), we perform simple arithmetic operations to solve it. Start by multiplying \( 0.5 \) by 18, resulting in 9. Then subtract this product from 205 to yield \( 196 \). This final value is the maximum heart rate for an 18-year-old.

Algebraic calculations like these involve straightforward steps: plug in numbers, perform operations, and then solve for the unknowns. These calculations are fundamental in solving many real-world problems, enabling us to find precise answers efficiently.