Problem 81 - College Algebra

Question

The optimum heart rate is the rate that a person should achieve during exercise for the exercise to be most beneficial. The algebraic expression $$ 0.6(220-a) $$ describes a person's optimum heart rate, in beats per minute, where \(a\) represents the age of the person. a. Use the distributive property to rewrite the algebraic expression without parentheses. b. Use each form of the algebraic expression to determine the optimum heart rate for a 20 -year-old runner.

Step-by-Step Solution

Answer

The distributed form of the algebraic expression is \(132-0.6a\), which outputs the same optimum heart rate as the original equation for a 20-year old runner, which is 120 beats per minute.

Step 1: Distribute the factor outside the parentheses

Using the distributive property, we can rewrite the given expression as \(0.6 \times 220 - 0.6 \times a\). After computing the products, the equation becomes \(132 - 0.6a\)

Step 2: Evaluate the optimum heart rate using the distributed form

Substitute \(a = 20\) into the distributed algebraic expression and evaluate. That gives \(132 - 0.6 \times 20\), which simplifies to \(132 - 12 = 120\) beats per minute

Step 3: Evaluate the optimum heart rate using the original form

Substitute \(a = 20\) into the original algebraic expression and evaluate. That gives \(0.6 \times (220 - 20) = 0.6 \times 200 = 120\) beats per minute. The results of both expressions are the same, confirming that distribution was done correctly.

Key Concepts

Optimum Heart Rate

Understanding the concept of optimum heart rate is crucial for anyone looking to get the most out of their workouts. The optimum heart rate helps ensure that exercise is effective and safe for your heart.

• Definition: The optimum heart rate is the ideal heart rate that one should aim for during exercise to gain maximum cardiovascular benefits.
• Importance: Maintaining this rate improves heart health, burns calories efficiently, and enhances fitness levels.
• Calculation: Often calculated using the formula: \(0.6 \times (220 - \text{age})\), where age is the individual's current age in years.

During exercise, consistently monitoring your heart rate against your calculated optimum heart rate can help guide the intensity of your workout. Staying within this heart rate zone not only maximizes physical benefits but also minimizes the risk of overexertion.

Algebraic Expressions

Algebraic expressions are the foundation of solving many mathematical problems, including those relating to real-life scenarios like calculating optimum heart rates.

• Components: An algebraic expression consists of numbers, variables, and operations. For example, \(0.6(220-a)\).
• Variables: They are symbols, like \(a\), that represent unknown or changeable values. Here, \(a\) stands for age.
• Operations: These include addition, subtraction, multiplication, and division.

The combinational structure of variables and constants in a single expression allows us to model real-world scenarios and solve for unknown quantities. The ability to manipulate these expressions using properties like distribution enhances our ability to solve complex problems.

Substitution Method

The substitution method is a powerful algebraic tool that helps solve expressions and equations efficiently. This method involves replacing variables with numbers to simplify expressions or solve equations.

• Process: Identify what the variable represents and substitute its value into the equation.
• Example: In our exercise, we substitute \(a = 20\) into both forms of the expression.
• Benefits: This method simplifies complex algebraic expressions and makes them manageable.

In the given exercise, substituting the age into the expressions helps verify the calculations. This ensures the solution is correct and demonstrates how changing one variable affects the outcome.