Write As A Power Function:$y$ Varies Inversely With The Cube Of $x$, And $y$ Is 27 When $x$ Is $\frac{2}{3}$.$y = \frac{k}{x^3}$Find $k$:$27 =

Introduction

In mathematics, inverse variation and power functions are two fundamental concepts that help us understand how different variables interact with each other. In this article, we will explore the relationship between a power function and inverse variation, and how to write a power function in the form of $y = \frac{k}{x^3}$.

What is Inverse Variation?

Inverse variation is a relationship between two variables where one variable increases as the other decreases, and vice versa. This relationship can be represented by the equation $y = \frac{k}{x^n}$, where $k$ is a constant and $n$ is a positive integer.

What is a Power Function?

A power function is a mathematical function that can be written in the form $y = ax^b$, where $a$ and $b$ are constants. Power functions are used to model a wide range of real-world phenomena, including population growth, chemical reactions, and electrical circuits.

The Relationship Between Inverse Variation and Power Functions

When we say that $y$ varies inversely with the cube of $x$, we mean that $y$ is equal to a constant $k$ divided by the cube of $x$. This relationship can be represented by the equation $y = \frac{k}{x^3}$.

Finding the Value of $k$

To find the value of $k$, we can use the given information that $y$ is 27 when $x$ is $\frac{2}{3}$. We can substitute these values into the equation $y = \frac{k}{x^3}$ and solve for $k$.

Substituting the Given Values

We are given that $y$ is 27 when $x$ is $\frac{2}{3}$. We can substitute these values into the equation $y = \frac{k}{x^3}$:

$$27 = \frac{k}{\left(\frac{2}{3}\right)^3}$$

Simplifying the Equation

To simplify the equation, we can raise $\frac{2}{3}$ to the power of 3:

$$27 = \frac{k}{\frac{8}{27}}$$

Multiplying Both Sides by $\frac{8}{27}$

To isolate $k$, we can multiply both sides of the equation by $\frac{8}{27}$:

$$k = 27 \times \frac{8}{27}$$

Simplifying the Equation

To simplify the equation, we can cancel out the common factor of 27:

$$k = 8$$

Conclusion

In this article, we have explored the relationship between inverse variation and power functions. We have seen how to write a power function in the form of $y = \frac{k}{x^3}$, and how to find the value of $k$ using the given information. We have also seen how to simplify the equation and isolate $k$. With this knowledge, we can now write a power function in the form of $y = \frac{k}{x^3}$ and find the value of $k$ using the given information.

Example Use Cases

Inverse variation and power functions have many real-world applications. Here are a few examples:

• Physics: Inverse variation is used to model the relationship between the force of gravity and the distance between two objects.
• Chemistry: Power functions are used to model the relationship between the concentration of a chemical solution and the time it takes to reach equilibrium.
• Economics: Inverse variation is used to model the relationship between the price of a good and the quantity demanded.

Conclusion

In conclusion, inverse variation and power functions are two fundamental concepts in mathematics that help us understand how different variables interact with each other. By writing a power function in the form of $y = \frac{k}{x^3}$, we can model a wide range of real-world phenomena. With this knowledge, we can now write a power function in the form of $y = \frac{k}{x^3}$ and find the value of $k$ using the given information.

References

• [1]: "Inverse Variation and Power Functions" by [Author's Name]
• [2]: "Mathematics for Economists" by [Author's Name]
• [3]: "Physics for Scientists and Engineers" by [Author's Name]

Glossary

• Inverse Variation: A relationship between two variables where one variable increases as the other decreases, and vice versa.
• Power Function: A mathematical function that can be written in the form $y = ax^b$, where $a$ and $b$ are constants.
• Constant: A value that does not change.
• Integer: A whole number, either positive, negative, or zero.

Further Reading

For further reading on inverse variation and power functions, we recommend the following resources:

• [1]: "Inverse Variation and Power Functions" by [Author's Name]
• [2]: "Mathematics for Economists" by [Author's Name]
• [3]: "Physics for Scientists and Engineers" by [Author's Name]

FAQs

• Q: What is inverse variation?
• A: Inverse variation is a relationship between two variables where one variable increases as the other decreases, and vice versa.
• Q: What is a power function?
• A: A power function is a mathematical function that can be written in the form $y = ax^b$, where $a$ and $b$ are constants.
• Q: How do I write a power function in the form of $y = \frac{k}{x^3}$?
• A: To write a power function in the form of $y = \frac{k}{x^3}$, you need to substitute the given values into the equation and solve for $k$.
  Inverse Variation and Power Functions: A Q&A Article =====================================================

Introduction

Inverse variation and power functions are two fundamental concepts in mathematics that help us understand how different variables interact with each other. In this article, we will answer some of the most frequently asked questions about inverse variation and power functions.

Q: What is inverse variation?

A: Inverse variation is a relationship between two variables where one variable increases as the other decreases, and vice versa. This relationship can be represented by the equation $y = \frac{k}{x^n}$, where $k$ is a constant and $n$ is a positive integer.

Q: What is a power function?

A: A power function is a mathematical function that can be written in the form $y = ax^b$, where $a$ and $b$ are constants. Power functions are used to model a wide range of real-world phenomena, including population growth, chemical reactions, and electrical circuits.

Q: How do I write a power function in the form of $y = \frac{k}{x^3}$?

A: To write a power function in the form of $y = \frac{k}{x^3}$, you need to substitute the given values into the equation and solve for $k$. Here's an example:

Suppose we are given that $y$ is 27 when $x$ is $\frac{2}{3}$. We can substitute these values into the equation $y = \frac{k}{x^3}$:

$$27 = \frac{k}{\left(\frac{2}{3}\right)^3}$$

To simplify the equation, we can raise $\frac{2}{3}$ to the power of 3:

$$27 = \frac{k}{\frac{8}{27}}$$

To isolate $k$, we can multiply both sides of the equation by $\frac{8}{27}$:

$$k = 27 \times \frac{8}{27}$$

To simplify the equation, we can cancel out the common factor of 27:

$$k = 8$$

Q: What is the difference between inverse variation and power functions?

A: Inverse variation and power functions are two related but distinct concepts. Inverse variation is a relationship between two variables where one variable increases as the other decreases, and vice versa. Power functions, on the other hand, are mathematical functions that can be written in the form $y = ax^b$, where $a$ and $b$ are constants.

Q: How do I use inverse variation and power functions in real-world applications?

A: Inverse variation and power functions have many real-world applications. Here are a few examples:

• Physics: Inverse variation is used to model the relationship between the force of gravity and the distance between two objects.
• Chemistry: Power functions are used to model the relationship between the concentration of a chemical solution and the time it takes to reach equilibrium.
• Economics: Inverse variation is used to model the relationship between the price of a good and the quantity demanded.

Q: What are some common mistakes to avoid when working with inverse variation and power functions?

A: Here are a few common mistakes to avoid when working with inverse variation and power functions:

• Not checking the units: Make sure to check the units of the variables in the equation to ensure that they are consistent.
• Not simplifying the equation: Make sure to simplify the equation as much as possible to avoid unnecessary complexity.
• Not using the correct formula: Make sure to use the correct formula for the problem at hand.

Q: How do I choose the correct formula for a problem involving inverse variation and power functions?

A: To choose the correct formula for a problem involving inverse variation and power functions, you need to consider the following factors:

• The type of relationship: Is the relationship between the variables direct or inverse?
• The power of the variable: Is the variable raised to a power or not?
• The units of the variables: Are the units of the variables consistent?

By considering these factors, you can choose the correct formula for the problem at hand.

Conclusion

In conclusion, inverse variation and power functions are two fundamental concepts in mathematics that help us understand how different variables interact with each other. By understanding these concepts and how to apply them in real-world applications, you can solve a wide range of problems involving inverse variation and power functions.

References

• [1]: "Inverse Variation and Power Functions" by [Author's Name]
• [2]: "Mathematics for Economists" by [Author's Name]
• [3]: "Physics for Scientists and Engineers" by [Author's Name]

Glossary

• Inverse Variation: A relationship between two variables where one variable increases as the other decreases, and vice versa.
• Power Function: A mathematical function that can be written in the form $y = ax^b$, where $a$ and $b$ are constants.
• Constant: A value that does not change.
• Integer: A whole number, either positive, negative, or zero.

Further Reading

For further reading on inverse variation and power functions, we recommend the following resources:

• [1]: "Inverse Variation and Power Functions" by [Author's Name]
• [2]: "Mathematics for Economists" by [Author's Name]
• [3]: "Physics for Scientists and Engineers" by [Author's Name]

FAQs

• Q: What is inverse variation?
• A: Inverse variation is a relationship between two variables where one variable increases as the other decreases, and vice versa.
• Q: What is a power function?
• A: A power function is a mathematical function that can be written in the form $y = ax^b$, where $a$ and $b$ are constants.
• Q: How do I write a power function in the form of $y = \frac{k}{x^3}$?
• A: To write a power function in the form of $y = \frac{k}{x^3}$, you need to substitute the given values into the equation and solve for $k$.