Write An Equation Describing The Relationship Of The Given Variables And Solve For $y$.$y$ Varies Inversely With The Cube Root Of $x$. When $x = 27$, $y = 5$. Find $y$ When $x = 125$.The

Understanding Inverse Variation

Inverse variation is a mathematical concept where two variables are related in such a way that as one variable increases, the other decreases, and vice versa. This relationship can be described using the equation $y = \frac{k}{x}$, where $y$ is the dependent variable, $x$ is the independent variable, and $k$ is a constant.

The Relationship Between $y$ and the Cube Root of $x$

In this problem, we are given that $y$ varies inversely with the cube root of $x$. This means that the relationship between $y$ and $x$ can be described using the equation $y = \frac{k}{\sqrt[3]{x}}$. We are also given that when $x = 27$, $y = 5$. We can use this information to find the value of the constant $k$.

Finding the Value of $k$

To find the value of $k$, we can substitute the given values of $x$ and $y$ into the equation $y = \frac{k}{\sqrt[3]{x}}$. This gives us:

$$5 = \frac{k}{\sqrt[3]{27}}$$

Since $\sqrt[3]{27} = 3$, we can simplify the equation to:

$$5 = \frac{k}{3}$$

To solve for $k$, we can multiply both sides of the equation by 3:

$$k = 15$$

The Equation Describing the Relationship Between $y$ and $x$

Now that we have found the value of $k$, we can write the equation describing the relationship between $y$ and $x$:

$$y = \frac{15}{\sqrt[3]{x}}$$

Solving for $y$ When $x = 125$

We are asked to find the value of $y$ when $x = 125$. We can substitute this value of $x$ into the equation:

$$y = \frac{15}{\sqrt[3]{125}}$$

Since $\sqrt[3]{125} = 5$, we can simplify the equation to:

$$y = \frac{15}{5}$$

To solve for $y$, we can divide both sides of the equation by 5:

$$y = 3$$

Conclusion

In this problem, we have described the relationship between $y$ and the cube root of $x$ using the equation $y = \frac{k}{\sqrt[3]{x}}$. We have found the value of the constant $k$ by substituting the given values of $x$ and $y$ into the equation. We have then used this value of $k$ to write the equation describing the relationship between $y$ and $x$. Finally, we have solved for $y$ when $x = 125$.

Mathematical Formulas and Equations

• $y = \frac{k}{\sqrt[3]{x}}$
• $5 = \frac{k}{\sqrt[3]{27}}$
• $k = 15$
• $y = \frac{15}{\sqrt[3]{x}}$
• $y = \frac{15}{5}$
• $y = 3$

Key Terms and Concepts

• Inverse variation
• Relationship between variables
• Constant $k$
• Cube root of $x$
• Equation $y = \frac{k}{\sqrt[3]{x}}$
• Solving for $y$ when $x = 125$
  Inverse Variation Q&A =========================

Frequently Asked Questions About Inverse Variation

Inverse variation is a mathematical concept that can be a bit tricky to understand. In this article, we will answer some of the most frequently asked questions about inverse variation.

Q: What is inverse variation?

A: Inverse variation is a mathematical concept where two variables are related in such a way that as one variable increases, the other decreases, and vice versa. This relationship can be described using the equation $y = \frac{k}{x}$, where $y$ is the dependent variable, $x$ is the independent variable, and $k$ is a constant.

Q: What is the difference between direct and inverse variation?

A: Direct variation is a mathematical concept where two variables are related in such a way that as one variable increases, the other also increases. Inverse variation, on the other hand, is a mathematical concept where two variables are related in such a way that as one variable increases, the other decreases, and vice versa.

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

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

1. Graph the data points on a coordinate plane.
2. If the graph is a hyperbola, then the relationship is inverse variation.
3. If the graph is a straight line, then the relationship is direct variation.

Q: What is the equation for inverse variation?

A: The equation for inverse variation is $y = \frac{k}{x}$, where $y$ is the dependent variable, $x$ is the independent variable, and $k$ is a constant.

Q: How do I find the value of $k$ in an inverse variation equation?

A: To find the value of $k$ in an inverse variation equation, you can use the following steps:

1. Substitute the given values of $x$ and $y$ into the equation.
2. Solve for $k$.

Q: What is the relationship between $y$ and the cube root of $x$ in an inverse variation equation?

A: In an inverse variation equation, the relationship between $y$ and the cube root of $x$ is described using the equation $y = \frac{k}{\sqrt[3]{x}}$.

Q: How do I solve for $y$ in an inverse variation equation?

A: To solve for $y$ in an inverse variation equation, you can use the following steps:

1. Substitute the given values of $x$ and $k$ into the equation.
2. Solve for $y$.

Q: What is the significance of inverse variation in real-life situations?

A: Inverse variation is significant in real-life situations such as:

• Physics: Inverse variation is used to describe the relationship between the force of gravity and the distance between two objects.
• Economics: Inverse variation is used to describe the relationship between the price of a product and the quantity demanded.
• Engineering: Inverse variation is used to describe the relationship between the speed of a vehicle and the distance traveled.

Conclusion

Inverse variation is a mathematical concept that can be used to describe the relationship between two variables. In this article, we have answered some of the most frequently asked questions about inverse variation. We hope that this article has provided you with a better understanding of inverse variation and its applications.

Mathematical Formulas and Equations

• $y = \frac{k}{x}$
• $y = \frac{k}{\sqrt[3]{x}}$
• $k = \frac{y \cdot x}{1}$

Key Terms and Concepts

• Inverse variation
• Relationship between variables
• Constant $k$
• Cube root of $x$
• Equation $y = \frac{k}{x}$
• Solving for $y$ in an inverse variation equation