The Equation with the Rate

Oct 13, 2025 by miodeya.com 106 views

Introduction

In this article, we will explore the relationship between time and distance, specifically in the context of Jeff's hike. We will examine the equation that represents the relationship between the time, $t$ , in hours he hikes and the distance, $d$ , in miles. Additionally, we will discuss whether the graph of this relationship will be continuous or discrete.

The Problem

Jeff hiked for 2 hours and traveled 5 miles. If he continues at the same pace, which equation will show the relationship between the time, $t$ , in hours he hikes to distance, $d$ , in miles?

The Equation

To find the equation that represents the relationship between time and distance, we need to use the concept of rate and proportion. Since Jeff continues at the same pace, we can assume that his rate of travel remains constant. We can use the formula:

d = rt

where $d$ is the distance traveled, $r$ is the rate of travel, and $t$ is the time traveled.

Finding the Rate

We are given that Jeff hiked for 2 hours and traveled 5 miles. We can use this information to find his rate of travel. We can rearrange the formula to solve for $r$ :

r = \frac{d}{t}

Substituting the given values, we get:

r = \frac{5}{2}

So, Jeff's rate of travel is 2.5 miles per hour.

The Equation with the Rate

Now that we have found Jeff's rate of travel, we can substitute it into the original equation:

d = rt

d = 2.5t

This is the equation that represents the relationship between time and distance.

Graphing the Relationship

The graph of this relationship will be a straight line, since the equation is linear. The graph will be continuous, since the relationship between time and distance is smooth and continuous.

Conclusion

In conclusion, the equation that represents the relationship between time and distance is $d = 2.5t$ . The graph of this relationship will be a continuous straight line.

Continuous vs. Discrete Graphs

A continuous graph is one in which the relationship between the variables is smooth and continuous. In this case, the relationship between time and distance is smooth and continuous, since Jeff's rate of travel remains constant.

On the other hand, a discrete graph is one in which the relationship between the variables is not smooth and continuous. For example, if we were to graph the number of hours Jeff hikes against the number of miles he travels, the graph would be discrete, since the number of hours and miles are both discrete variables.

Real-World Applications

The concept of rate and proportion has many real-world applications. For example, in finance, the rate of return on an investment is often used to calculate the future value of the investment. In physics, the rate of change of velocity is used to calculate the acceleration of an object.

Example Problems

If Jeff hikes for 4 hours, how many miles will he travel?
If Jeff hikes for 6 hours, how many miles will he travel?

Solutions

We can use the equation $d = 2.5t$ to solve for $d$ :

d = 2.5(4)

d = 10

So, Jeff will travel 10 miles in 4 hours.

We can use the equation $d = 2.5t$ to solve for $d$ :

d = 2.5(6)

d = 15

So, Jeff will travel 15 miles in 6 hours.

Conclusion

Introduction

In our previous article, we explored the relationship between time and distance, specifically in the context of Jeff's hike. We examined the equation that represents the relationship between the time, $t$ , in hours he hikes and the distance, $d$ , in miles. In this article, we will answer some frequently asked questions about this relationship.

Q: What is the equation that represents the relationship between time and distance?

A: The equation that represents the relationship between time and distance is $d = 2.5t$ . This equation shows that the distance traveled is directly proportional to the time traveled.

Q: What is the rate of travel?

A: The rate of travel is the distance traveled per unit of time. In this case, the rate of travel is 2.5 miles per hour.

Q: Is the graph of this relationship continuous or discrete?

A: The graph of this relationship is continuous, since the relationship between time and distance is smooth and continuous.

Q: What are some real-world applications of this concept?

A: The concept of rate and proportion has many real-world applications, including finance, physics, and engineering. For example, in finance, the rate of return on an investment is often used to calculate the future value of the investment. In physics, the rate of change of velocity is used to calculate the acceleration of an object.

Q: How can I use this concept to solve problems in mathematics and other fields?

A: You can use this concept to solve problems in mathematics and other fields by applying the formula $d = rt$ . This formula can be used to calculate the distance traveled, given the rate of travel and the time traveled.

Q: What are some example problems that I can use to practice this concept?

A: Here are some example problems that you can use to practice this concept:

If a car travels at a rate of 60 miles per hour, how far will it travel in 2 hours?
If a person hikes at a rate of 3 miles per hour, how far will they travel in 4 hours?
If a train travels at a rate of 80 miles per hour, how far will it travel in 5 hours?

Solutions

We can use the formula $d = rt$ to solve for $d$ :

d = 60(2)

d = 120

So, the car will travel 120 miles in 2 hours.

We can use the formula $d = rt$ to solve for $d$ :

d = 3(4)

d = 12

So, the person will travel 12 miles in 4 hours.

We can use the formula $d = rt$ to solve for $d$ :

d = 80(5)

d = 400

So, the train will travel 400 miles in 5 hours.

Conclusion

In conclusion, the equation that represents the relationship between time and distance is $d = 2.5t$ . The graph of this relationship is continuous, and the concept of rate and proportion has many real-world applications. By applying the formula $d = rt$ , you can solve problems in mathematics and other fields.