Problem 64 - Algebra 1

Question

MARATHON In Exercises \(64-66,\) you are running in a marathon. You either run 8 miles per hour or walk 4 miles per hour. Write an equation to show the relationship between time run and time walked during the 26.2 -mile course.

Step-by-Step Solution

Answer

The relationship between the time spent running and the time spent walking during the 26.2-mile marathon is expressed by the equation \(8t_r + 4t_w = 26.2\).

Step 1: Determine the time for each action

Begin by applying the formula for time which is distance divided by speed. Call the time spent running as \(t_r\) and the time spent walking as \(t_w\). According to the problem, when you are running, you can cover 8 miles in one hour and when walking, you can cover 4 miles in one hour. Hence, \(t_r = d_r / 8\) and \(t_w = d_w / 4\).

Step 2: Form a relationship between running and walking distances

We know that the total distance of the marathon is 26.2 miles. Therefore, the distance covered running (\(d_r\)) plus the distance covered walking (\(d_w\)) should be equal to the total distance. Therefore, \(d_r + d_w = 26.2\) miles.

Step 3: Insert the Distance-Time relationship into the Distance Equation

Substitute the equations from Step 1 for \(d_r\) and \(d_w\) into the equation from Step 2. This gives: \(8t_r + 4t_w = 26.2\). This is the final relationship between the times spent running and walking during the marathon.

Key Concepts

Speed and Distance Formula

Understanding the speed and distance formula is key when analyzing any situation involving motion, like a marathon. The foundational formula to grasp here is:

• Speed = Distance / Time

This tells us how fast something is moving through space over a certain duration.
To find how long an activity (running or walking) takes, we can rearrange the formula to:

• Time = Distance / Speed

This is crucial. Knowing the speed will help determine the time taken to cover a set distance.
In the context of the marathon exercise, the runner moves at 8 miles per hour and walks at 4 miles per hour. By applying the formula, you calculate how much time is spent on each action. This is shown as:

• Running time: \( t_r = \frac{d_r}{8} \)
• Walking time: \( t_w = \frac{d_w}{4} \)

These equations represent the simple yet powerful connection between speed, distance, and time.

Linear Equations

Linear equations come into play by providing a relationship between different variables. In situations involving constant speeds and distances, linear equations can succinctly model behavior.
The general form of a linear equation is \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants, and \( x \), \( y \) are variables. This kind of equation helps describe straight-line relationships.
In the marathon problem, the equation \( 8t_r + 4t_w = 26.2 \) represents a linear relationship. The elements:

• \( 8t_r \): time spent running with speed 8 mph.
• \( 4t_w \): time spent walking with speed 4 mph.
• 26.2: the total distance of the marathon.

This linear equation efficiently ties together the task of running and walking in a simple, understandable manner. In reality, any adjustments in time or distance would reflect as a shift or change along this equation, emphasizing its linear trait.

Problem-Solving in Algebra

Problem-solving in algebra is a skill that involves understanding a problem, devising a plan, executing that plan, and then evaluating the solution.
In the context of our marathon exercise, there are several steps to follow:

• Understand the problem: Break down what is given—the speeds, the total distance, and what needs to be found: the relationship between running and walking times.
• Devise a plan: Use known formulas, like the speed and distance formula, to express the times spent in each activity in terms of distances covered.
• Execute the plan: Set up the relationship between time and distance and insert these into a linear equation.
• Evaluate the solution: Check if the solution makes sense in the context. Verify if the total distance matches 26.2 miles.

Practicing these steps enhances problem-solving skills by applying logical reasoning and mathematical tools like formulas and equations to find solutions efficiently.