If Y Y Y Varies Directly As X X X , And Y Y Y Is 20 When X X X Is 4, What Is The Constant Of Variation For This Relation?A. 1 5 \frac{1}{5} 5 1 ​ B. 4 5 \frac{4}{5} 5 4 ​ C. 5 D. 16

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Direct variation is a type of relationship between two variables, where one variable is a constant multiple of the other. In other words, as one variable increases or decreases, the other variable increases or decreases at a constant rate. This relationship can be represented by the equation y = kx, where y is the dependent variable, x is the independent variable, and k is the constant of variation.

The Constant of Variation

The constant of variation, denoted by k, is a measure of the rate at which the dependent variable changes in response to changes in the independent variable. In other words, it represents the slope of the line that represents the direct variation relationship. To find the constant of variation, we can use the formula k = y/x, where y is the value of the dependent variable and x is the value of the independent variable.

Example Problem

If y varies directly as x, and y is 20 when x is 4, what is the constant of variation for this relation?

Step 1: Write the equation of direct variation

Since y varies directly as x, we can write the equation of direct variation as y = kx.

Step 2: Plug in the given values

We are given that y is 20 when x is 4. We can plug these values into the equation of direct variation to get:

20 = k(4)

Step 3: Solve for k

To solve for k, we can divide both sides of the equation by 4:

k = 20/4 k = 5

Conclusion

Therefore, the constant of variation for this relation is 5.

Why is this important?

Understanding direct variation and the constant of variation is important in many real-world applications, such as physics, engineering, and economics. For example, the constant of variation can be used to model the relationship between the distance traveled by an object and the time it takes to travel that distance.

Real-World Applications

Direct variation has many real-world applications, including:

  • Physics: The constant of variation can be used to model the relationship between the distance traveled by an object and the time it takes to travel that distance.
  • Engineering: The constant of variation can be used to model the relationship between the voltage and current in an electrical circuit.
  • Economics: The constant of variation can be used to model the relationship between the price of a good and the quantity demanded.

Conclusion

In conclusion, direct variation is a type of relationship between two variables, where one variable is a constant multiple of the other. The constant of variation, denoted by k, is a measure of the rate at which the dependent variable changes in response to changes in the independent variable. By understanding direct variation and the constant of variation, we can model many real-world relationships and make predictions about future behavior.

Key Takeaways

  • Direct variation is a type of relationship between two variables, where one variable is a constant multiple of the other.
  • The constant of variation, denoted by k, is a measure of the rate at which the dependent variable changes in response to changes in the independent variable.
  • The constant of variation can be found using the formula k = y/x.
  • Direct variation has many real-world applications, including physics, engineering, and economics.

References

Q: What is direct variation?

A: Direct variation is a type of relationship between two variables, where one variable is a constant multiple of the other. In other words, as one variable increases or decreases, the other variable increases or decreases at a constant rate.

Q: How is direct variation represented mathematically?

A: Direct variation is represented mathematically by the equation y = kx, where y is the dependent variable, x is the independent variable, and k is the constant of variation.

Q: What is the constant of variation?

A: The constant of variation, denoted by k, is a measure of the rate at which the dependent variable changes in response to changes in the independent variable.

Q: How do I find the constant of variation?

A: To find the constant of variation, you can use the formula k = y/x, where y is the value of the dependent variable and x is the value of the independent variable.

Q: What is an example of direct variation in real life?

A: An example of direct variation in real life is the relationship between the distance traveled by a car and the time it takes to travel that distance. As the distance traveled increases, the time it takes to travel that distance also increases at a constant rate.

Q: Can you give me another example of direct variation?

A: Another example of direct variation is the relationship between the voltage and current in an electrical circuit. As the voltage increases, the current also increases at a constant rate.

Q: How do I determine if a relationship is a direct variation?

A: To determine if a relationship is a direct variation, you can use the following steps:

  1. Plot the data on a graph.
  2. Check if the graph is a straight line.
  3. If the graph is a straight line, then the relationship is a direct variation.

Q: What are some common applications of direct variation?

A: Some common applications of direct variation include:

  • Physics: Direct variation is used to model the relationship between the distance traveled by an object and the time it takes to travel that distance.
  • Engineering: Direct variation is used to model the relationship between the voltage and current in an electrical circuit.
  • Economics: Direct variation is used to model the relationship between the price of a good and the quantity demanded.

Q: Can you give me some practice problems to try?

A: Here are some practice problems to try:

  1. If y varies directly as x, and y is 30 when x is 5, what is the constant of variation?
  2. If the distance traveled by a car varies directly as the time it takes to travel that distance, and the car travels 200 miles in 4 hours, what is the constant of variation?
  3. If the voltage in an electrical circuit varies directly as the current, and the voltage is 12 volts when the current is 2 amps, what is the constant of variation?

Q: How do I solve these practice problems?

A: To solve these practice problems, you can use the following steps:

  1. Write the equation of direct variation (y = kx).
  2. Plug in the given values into the equation.
  3. Solve for k (the constant of variation).

Q: What if I get stuck on a problem?

A: If you get stuck on a problem, try the following:

  1. Read the problem carefully and make sure you understand what is being asked.
  2. Break down the problem into smaller steps.
  3. Use a diagram or graph to help you visualize the problem.
  4. Ask a teacher or tutor for help.

Conclusion

Direct variation is a type of relationship between two variables, where one variable is a constant multiple of the other. By understanding direct variation and the constant of variation, you can model many real-world relationships and make predictions about future behavior.