Carlos Begins His Workout By Stretching. Then He Spends An Equal Amount Of Time On Each Workout Machine. The Equation $y = 8x + 10$ Represents The Total Time, In Minutes, Of Carlos' Workout If He Uses $x$ Workout Machines.Explain What

Feb 27, 2025 by ADMIN 235 views

**Understanding the Relationship Between Workout Machines and Total Time**

Introduction

Carlos is a fitness enthusiast who takes his workouts seriously. He begins each session by stretching, and then he spends an equal amount of time on each workout machine. The equation $y = 8x + 10$ represents the total time, in minutes, of Carlos' workout if he uses $x$ workout machines. In this article, we will explore the relationship between the number of workout machines and the total time spent on a workout.

The Equation

The equation $y = 8x + 10$ is a linear equation that represents the total time, in minutes, of Carlos' workout. The variable $x$ represents the number of workout machines used, and the variable $y$ represents the total time spent on the workout. The equation can be broken down into two parts:

The slope-intercept form of the equation is $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept. In this case, the slope is $8$ and the y-intercept is $10$ .
The equation can be rewritten as $y = 8x + 10$ , where $8x$ represents the time spent on the workout machines and $10$ represents the time spent on stretching.

Interpreting the Equation

To understand the relationship between the number of workout machines and the total time spent on a workout, we need to analyze the equation. The slope of the equation is $8$ , which means that for every additional workout machine used, the total time spent on the workout increases by $8$ minutes. This is because each workout machine takes an equal amount of time, and the total time is the sum of the time spent on each machine.

Example

Let's say Carlos wants to use $3$ workout machines. To find the total time spent on the workout, we can plug in $x = 3$ into the equation:

y = 8(3) + 10

y = 24 + 10

y = 34

Therefore, if Carlos uses $3$ workout machines, the total time spent on the workout is $34$ minutes.

Graphing the Equation

To visualize the relationship between the number of workout machines and the total time spent on a workout, we can graph the equation. The graph of the equation is a straight line with a slope of $8$ and a y-intercept of $10$ . The x-axis represents the number of workout machines used, and the y-axis represents the total time spent on the workout.

Conclusion

In conclusion, the equation $y = 8x + 10$ represents the total time, in minutes, of Carlos' workout if he uses $x$ workout machines. The slope of the equation is $8$ , which means that for every additional workout machine used, the total time spent on the workout increases by $8$ minutes. By analyzing the equation and graphing it, we can understand the relationship between the number of workout machines and the total time spent on a workout.

The Importance of Stretching

Carlos begins each workout by stretching, and then he spends an equal amount of time on each workout machine. The time spent on stretching is represented by the y-intercept of the equation, which is $10$ minutes. This means that regardless of the number of workout machines used, Carlos always spends $10$ minutes stretching.

The Impact of Using More Workout Machines

The equation $y = 8x + 10$ shows that the total time spent on the workout increases by $8$ minutes for every additional workout machine used. This means that if Carlos uses more workout machines, the total time spent on the workout will increase. For example, if Carlos uses $4$ workout machines, the total time spent on the workout will be:

y = 8(4) + 10

y = 32 + 10

y = 42

Therefore, if Carlos uses $4$ workout machines, the total time spent on the workout is $42$ minutes.

From a mathematical perspective, the equation $y = 8x + 10$ represents a linear relationship between the number of workout machines and the total time spent on a workout. The slope of the equation is $8$ , which means that the total time spent on the workout increases by $8$ minutes for every additional workout machine used.

The Implications of the Equation

The equation $y = 8x + 10$ has several implications for Carlos' workout routine. For example:

If Carlos wants to reduce the total time spent on the workout, he can use fewer workout machines.
If Carlos wants to increase the total time spent on the workout, he can use more workout machines.
If Carlos wants to maintain the same total time spent on the workout, he can adjust the number of workout machines used accordingly.

Conclusion

The equation $y = 8x + 10$ has several real-world applications in the fitness industry. For example:

Personal trainers can use the equation to plan workouts for their clients based on the number of machines available.
Fitness centers can use the equation to determine the optimal number of machines to purchase based on the expected demand.
Athletes can use the equation to plan their workouts and adjust the number of machines used accordingly.

Conclusion

Frequently Asked Questions

Q: What is the equation $y = 8x + 10$ and how does it relate to Carlos' workout? A: The equation $y = 8x + 10$ represents the total time, in minutes, of Carlos' workout if he uses $x$ workout machines. The equation shows that for every additional workout machine used, the total time spent on the workout increases by $8$ minutes.

Q: What is the slope of the equation and how does it affect the total time spent on the workout? A: The slope of the equation is $8$ , which means that for every additional workout machine used, the total time spent on the workout increases by $8$ minutes.

Q: How does the number of workout machines affect the total time spent on the workout? A: The number of workout machines used directly affects the total time spent on the workout. For every additional workout machine used, the total time spent on the workout increases by $8$ minutes.

Q: What is the y-intercept of the equation and how does it relate to Carlos' workout? A: The y-intercept of the equation is $10$ , which represents the time spent on stretching. Regardless of the number of workout machines used, Carlos always spends $10$ minutes stretching.

Q: Can you provide an example of how to use the equation to find the total time spent on the workout? A: Let's say Carlos wants to use $3$ workout machines. To find the total time spent on the workout, we can plug in $x = 3$ into the equation:

y = 8(3) + 10

y = 24 + 10

y = 34

Therefore, if Carlos uses $3$ workout machines, the total time spent on the workout is $34$ minutes.

Q: How can the equation be used in real-world applications? A: The equation $y = 8x + 10$ has several real-world applications in the fitness industry. For example, personal trainers can use the equation to plan workouts for their clients based on the number of machines available, and fitness centers can use the equation to determine the optimal number of machines to purchase based on the expected demand.

Q: What are some implications of the equation for Carlos' workout routine? A: The equation $y = 8x + 10$ has several implications for Carlos' workout routine. For example, if Carlos wants to reduce the total time spent on the workout, he can use fewer workout machines, and if Carlos wants to increase the total time spent on the workout, he can use more workout machines.

Q: Can you provide a graph of the equation to visualize the relationship between the number of workout machines and the total time spent on the workout? A: Yes, the graph of the equation is a straight line with a slope of $8$ and a y-intercept of $10$ . The x-axis represents the number of workout machines used, and the y-axis represents the total time spent on the workout.

Conclusion

Additional Resources

For more information on the equation $y = 8x + 10$ , please refer to the previous article.
For more information on the relationship between workout machines and total time, please refer to the previous article.
For more information on real-world applications of the equation, please refer to the previous article.