Defining the Variables and Identifying the Rate

Oct 13, 2025 by recoveryshake.com 28 views

When a person travels from their home to their office, the total distance traveled can be modeled using a linear equation. In this scenario, we are given that the person has already traveled 3.5 miles and can cover 100 miles every 2 hours. To find the equation of a line that models the total miles traveled, we need to understand the relationship between the distance traveled and the time taken.

Defining the Variables and Identifying the Rate

Let's define the variables:

Distance traveled (d): The total distance traveled by the person in miles.
Time (t): The time taken by the person to travel the distance in hours.

We are given that the person can cover 100 miles every 2 hours. This means that the rate at which the person travels is 100 miles / 2 hours = 50 miles per hour.

Writing the Equation of the Line

The equation of a line that models the total miles traveled can be written in the form:

d = mt + b

where:

d: The total distance traveled in miles.
m: The rate at which the person travels in miles per hour.
t: The time taken by the person to travel the distance in hours.
b: The initial distance traveled by the person.

In this scenario, the initial distance traveled by the person is 3.5 miles. Therefore, the equation of the line can be written as:

d = 50t + 3.5

Interpreting the Equation

The equation d = 50t + 3.5 represents a linear relationship between the distance traveled and the time taken. The rate at which the person travels is 50 miles per hour, and the initial distance traveled is 3.5 miles.

Solving for Time

To find the time taken by the person to travel a certain distance, we can rearrange the equation to solve for t:

t = (d - 3.5) / 50

Example

Suppose the person wants to travel 150 miles. How long will it take them to reach their destination?

Using the equation t = (d - 3.5) / 50, we can substitute d = 150 and solve for t:

t = (150 - 3.5) / 50 t = 146.5 / 50 t = 2.93 hours

Therefore, it will take the person approximately 2.93 hours to travel 150 miles.

Conclusion

In this article, we have discussed how to find the equation of a line that models the total miles traveled by a person traveling from their home to their office. We have defined the variables, identified the rate, and written the equation of the line. We have also interpreted the equation and solved for time. By using this equation, we can determine the time taken by the person to travel a certain distance.

Key Takeaways

The equation of a line that models the total miles traveled is d = mt + b.
The rate at which the person travels is 50 miles per hour.
The initial distance traveled by the person is 3.5 miles.
The equation can be used to solve for time by rearranging it to t = (d - 3.5) / 50.

Final Thoughts

In our previous article, we discussed how to find the equation of a line that models the total miles traveled by a person traveling from their home to their office. We defined the variables, identified the rate, and wrote the equation of the line. In this article, we will answer some frequently asked questions related to the equation of a line that models the total miles traveled.

Q: What is the significance of the rate in the equation?

A: The rate in the equation represents the speed at which the person travels. In this scenario, the rate is 50 miles per hour, which means that the person can cover 50 miles in 1 hour.

Q: How do I determine the initial distance traveled by the person?

A: The initial distance traveled by the person is the distance traveled before the person starts traveling at the given rate. In this scenario, the initial distance traveled is 3.5 miles.

Q: Can I use the equation to find the distance traveled if I know the time taken?

A: Yes, you can use the equation to find the distance traveled if you know the time taken. To do this, you can rearrange the equation to solve for d:

d = 50t + 3.5

Q: What if the person travels at a different rate? How do I adjust the equation?

A: If the person travels at a different rate, you need to adjust the equation accordingly. For example, if the person travels at a rate of 75 miles per hour, the equation would be:

d = 75t + 3.5

Q: Can I use the equation to find the time taken if I know the distance traveled?

A: Yes, you can use the equation to find the time taken if you know the distance traveled. To do this, you can rearrange the equation to solve for t:

t = (d - 3.5) / 50

Q: What if the person travels a different initial distance? How do I adjust the equation?

A: If the person travels a different initial distance, you need to adjust the equation accordingly. For example, if the person travels an initial distance of 2 miles, the equation would be:

d = 50t + 2

Q: Can I use the equation to model the total miles traveled in a different scenario?

A: Yes, you can use the equation to model the total miles traveled in a different scenario. For example, you can use the equation to model the total miles traveled by a car traveling at a constant speed, or by a plane traveling at a constant speed.

Q: What are some real-world applications of the equation?

A: Some real-world applications of the equation include:

Calculating the time taken to travel a certain distance in a car, plane, or train.
Modeling the total miles traveled by a person or an object.
Determining the distance traveled by a person or an object given the time taken.
Calculating the speed of a person or an object given the distance traveled and the time taken.

Conclusion

In this article, we have answered some frequently asked questions related to the equation of a line that models the total miles traveled. We have discussed the significance of the rate in the equation, how to determine the initial distance traveled by the person, and how to adjust the equation for different scenarios. We have also provided some real-world applications of the equation. By understanding the equation and its applications, you can use it to model the total miles traveled in various scenarios.