Problem 126 - Introductory Algebra for College Students

Question

If your exercise goal is to improve overall health, the graph on the previous page shows the following range for target heart rate, \(H,\) in beats per minute: Lower limit of range $$H=\frac{1}{2}(220-a)$$ Upper limit of range $$H=\frac{3}{5}(220-a)$$ a. What is the lower limit of the heart range, in beats per minute, for a 30 -year-old with this exercise goal? b. What is the upper limit of the heart range, in beats per minute, for a 30 -year-old with this exercise goal?

Step-by-Step Solution

Answer

The lower limit of the heart rate range for a 30-year-old person is 95 beats per minute and the upper limit is 114 beats per minute.

Step 1: Calculate Lower Limit

First, insert the age of the person, in this case 30 years, into the equation for the lower limit of a heart rate. Therefore, the equation will be \(H=\frac{1}{2}(220-30)\).

Step 2: Solve Lower Limit

Perform the subtraction inside the brackets first due to the BIDMAS rule (Brackets, Indices, Division and Multiplication (from left to right), Addition and Subtraction (from left to right)). The equation will be \(H=\frac{1}{2} * 190\). Now, solve the multiplication, which will result in \(H=95\).

Step 3: Calculate Upper Limit

Next, insert the age of the person, again 30 years, into the equation for the upper limit of a heart rate. This will give the equation \(H=\frac{3}{5}(220-30)\).

Step 4: Solve Upper Limit

Using the BIDMAS rule, perform the subtraction inside the brackets first. The equation will be \(H=\frac{3}{5} * 190\). Finally, solve the multiplication which results in \(H=114\).

Key Concepts

Heart Rate Calculations

Understanding how to calculate the target heart rate is quite essential for anyone looking to manage their fitness levels effectively. The target heart rate is a way to calculate how fast your heart should be beating during exercise to be effective but safe as well. This rate varies for each person based on age and fitness level. In the exercise, two formulas are provided to represent the lower and upper limits of the heart rate range, represented as algebraic equations based on age. The aim is to work within this range to enhance cardiovascular health. To find these limits for a 30-year-old, replace "a" in the equations with the age of 30 years. It helps to put these calculations into context with a real-world scenario, such as exercising safely.

Algebraic Expressions

Algebraic expressions are fundamental in mathematics and are used to represent numbers symbolically allowing for various operations and problem-solving. In our exercise, we use the algebraic expressions to denote the formulas for the upper and lower heart rate limits. The expressions given are:

• Lower Limit: \(H=\frac{1}{2}(220-a)\)
• Upper Limit: \(H=\frac{3}{5}(220-a)\)

Here, "a" represents the age in years. By substituting age into these expressions, you are performing algebraic operations that simplify the mathematical statements into numerical values. In our problem, once substituting the age of 30, these expressions are simplified using basic arithmetic operations, elucidating clear numerical results.

BIDMAS Rule

BIDMAS (or PEMDAS) is an acronym that helps remember the order of operations in mathematical expressions. It stands for Brackets, Indices, Division and Multiplication, Addition and Subtraction. This is crucial when solving equations like the ones in our exercise because it ensures that calculations are done accurately and consistently. For instance, when solving the equation \(H=\frac{1}{2}(220-30)\), you need to resolve the brackets first, which involves the subtraction \((220-30)\). Only after that do you perform the multiplication \(\frac{1}{2} \times 190\). This rule is essential to avoid mistakes, especially in more complex algebraic expressions, ensuring each step is performed in the correct sequence.

Problem-Solving Steps

Efficient problem-solving involves following a structured approach to find solutions to given problems, particularly in mathematical contexts. Here are the steps used in our exercise: - Step 1: Identify and understand the problem, which means choosing the correct formula for each limit.
- Step 2: Substitute the given values, such as replacing the age in the formula.
- Step 3: Utilize the BIDMAS rule to solve the equations progressively, starting from solving within the brackets.
- Step 4: Complete the operations and provide final solutions.
By following these steps, problem-solving becomes more manageable, consistent, and less complicated. Each step brings you closer to the solution while ensuring that you thoroughly understand the process and logic behind each action taken.