After Running For 54 Minutes, She Completes 6 Kilometers. Her Trainer Writes An Equation Letting \[$ T \$\], The Time In Minutes, Represent The Independent Variable, And \[$ K \$\], The Number Of Kilometers, Represent The Dependent

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Introduction

In mathematics, relationships between variables are often represented using equations. These equations can be used to model real-world situations, such as the relationship between time and distance. In this article, we will explore the relationship between time and distance, using the example of a runner who completes 6 kilometers in 54 minutes.

The Problem

A runner completes 6 kilometers in 54 minutes. Her trainer wants to write an equation that represents the relationship between time and distance. The independent variable, { t $}$, represents the time in minutes, and the dependent variable, { k $}$, represents the number of kilometers.

Writing the Equation

To write the equation, we need to determine the relationship between time and distance. In this case, the runner completes 6 kilometers in 54 minutes. This means that the distance is directly proportional to the time. We can write the equation as:

{ k = \frac{6}{54}t $}$

This equation states that the number of kilometers, { k $}$, is equal to the product of the constant { \frac{6}{54} $}$ and the time, { t $}$.

Simplifying the Equation

We can simplify the equation by dividing both sides by { \frac{6}{54} $}$. This gives us:

{ k = \frac{1}{9}t $}$

This equation states that the number of kilometers, { k $}$, is equal to { \frac{1}{9} $}$ times the time, { t $}$.

Interpreting the Equation

The equation { k = \frac{1}{9}t $}$ tells us that for every minute the runner runs, she completes { \frac{1}{9} $}$ kilometers. This means that if the runner runs for 54 minutes, she will complete 6 kilometers.

Graphing the Equation

We can graph the equation { k = \frac{1}{9}t $}$ by plotting the points { (0, 0) $}$, { (54, 6) $}$, and { (108, 12) $}$. The resulting graph is a straight line with a slope of { \frac{1}{9} $}$.

Real-World Applications

The equation { k = \frac{1}{9}t $}$ has many real-world applications. For example, it can be used to model the relationship between time and distance for a runner, a car, or a plane. It can also be used to calculate the time it takes to complete a certain distance.

Conclusion

In conclusion, the equation { k = \frac{1}{9}t $}$ represents the relationship between time and distance. It tells us that for every minute the runner runs, she completes { \frac{1}{9} $}$ kilometers. This equation has many real-world applications and can be used to model the relationship between time and distance for a variety of situations.

Additional Examples

  • A car travels 120 kilometers in 2 hours. Write an equation that represents the relationship between time and distance.
  • A plane flies 500 kilometers in 1 hour. Write an equation that represents the relationship between time and distance.
  • A runner completes 10 kilometers in 60 minutes. Write an equation that represents the relationship between time and distance.

Solutions

  • A car travels 120 kilometers in 2 hours. Write an equation that represents the relationship between time and distance.

    { d = 60t $}$

  • A plane flies 500 kilometers in 1 hour. Write an equation that represents the relationship between time and distance.

    { d = 500t $}$

  • A runner completes 10 kilometers in 60 minutes. Write an equation that represents the relationship between time and distance.

    { d = \frac{5}{6}t $}$

Conclusion

Q: What is the relationship between time and distance?

A: The relationship between time and distance is represented by the equation { k = \frac{1}{9}t $}$, where { k $}$ is the number of kilometers and { t $}$ is the time in minutes.

Q: How do I use the equation to calculate the distance traveled?

A: To calculate the distance traveled, you can plug in the value of time into the equation { k = \frac{1}{9}t $}$. For example, if the runner runs for 54 minutes, you can calculate the distance traveled as follows:

{ k = \frac{1}{9} \times 54 $}$

{ k = 6 $}$

This means that the runner will travel 6 kilometers in 54 minutes.

Q: What if I want to calculate the time it takes to travel a certain distance?

A: To calculate the time it takes to travel a certain distance, you can rearrange the equation { k = \frac{1}{9}t $}$ to solve for time. This gives us:

{ t = 9k $}$

For example, if the runner wants to travel 10 kilometers, you can calculate the time it takes as follows:

{ t = 9 \times 10 $}$

{ t = 90 $}$

This means that the runner will take 90 minutes to travel 10 kilometers.

Q: Can I use the equation to model other real-world situations?

A: Yes, the equation { k = \frac{1}{9}t $}$ can be used to model other real-world situations where the relationship between time and distance is linear. For example, you can use the equation to model the relationship between time and distance for a car, a plane, or a train.

Q: What if the relationship between time and distance is not linear?

A: If the relationship between time and distance is not linear, you will need to use a different equation to model the situation. For example, if the relationship between time and distance is quadratic, you will need to use an equation of the form { k = at^2 $}$, where { a $}$ is a constant.

Q: Can I use the equation to solve problems involving multiple variables?

A: Yes, the equation { k = \frac{1}{9}t $}$ can be used to solve problems involving multiple variables. For example, you can use the equation to model the relationship between time, distance, and speed. This will require you to use a system of equations to solve the problem.

Q: Where can I find more information about the equation and its applications?

A: You can find more information about the equation and its applications in mathematics textbooks, online resources, and academic journals. You can also consult with a mathematics teacher or tutor for further guidance.

Conclusion

In conclusion, the equation { k = \frac{1}{9}t $}$ represents the relationship between time and distance. It can be used to model a variety of real-world situations, including the relationship between time and distance for a runner, a car, or a plane. The equation can also be used to solve problems involving multiple variables and can be applied to a wide range of fields, including physics, engineering, and economics.