Write An Equation Describing The Relationship Of The Given Variables And Solve For $y$.$y$ Varies Inversely With The Cube Root Of \$x$[/tex\]. When $x = 27$, Then $y = 5.5$. Find

Mar 1, 2025 by ADMIN 188 views

**Inverse Variation and the Relationship Between Variables**

Understanding Inverse Variation

Inverse variation is a mathematical relationship between two variables, where the product of the two variables remains constant. This relationship can be described by the equation: $y = \frac{k}{x^n}$, where $y$ is the dependent variable, $x$ is the independent variable, $k$ is a constant, and $n$ is the exponent that determines the type of variation.

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 by the equation: $y = \frac{k}{\sqrt[3]{x}}$, where $k$ is a constant.

Using the Given Values to Find the Constant $k$

We are given that when $x = 27$, then $y = 5.5$. We can use this information to find the value of the constant $k$. Substituting the given values into the equation, we get: $5.5 = \frac{k}{\sqrt[3]{27}}$.

Simplifying the Equation

To simplify the equation, we can evaluate the cube root of 27, which is equal to 3. Therefore, the equation becomes: $5.5 = \frac{k}{3}$.

Solving for $k$

To solve for $k$, we can multiply both sides of the equation by 3, which gives us: $k = 5.5 \times 3$.

Evaluating the Product

Evaluating the product, we get: $k = 16.5$.

The Final Equation

Now that we have found the value of the constant $k$, we can write the final equation that describes the relationship between $y$ and the cube root of $x$: $y = \frac{16.5}{\sqrt[3]{x}}$.

Solving for $y$

To solve for $y$, we can multiply both sides of the equation by the cube root of $x$, which gives us: $y = \frac{16.5}{x^{\frac{1}{3}}}$.

Simplifying the Equation

To simplify the equation, we can rewrite it as: $y = \frac{16.5}{\sqrt[3]{x}}$.

The Relationship Between $y$ and $x$

The final equation describes the relationship between $y$ and the cube root of $x$. This means that as the cube root of $x$ increases, the value of $y$ decreases, and vice versa.

Conclusion

In this problem, we have described the relationship between $y$ and the cube root of $x$ using the equation $y = \frac{16.5}{\sqrt[3]{x}}$. We have also solved for $y$ by multiplying both sides of the equation by the cube root of $x$. This equation can be used to find the value of $y$ for any given value of $x$.

Example Use Case

Suppose we want to find the value of $y$ when $x = 64$. We can substitute this value into the equation: $y = \frac{16.5}{\sqrt[3]{64}}$.

Evaluating the Cube Root

Evaluating the cube root of 64, we get: $\sqrt[3]{64} = 4$.

Substituting the Value

Substituting this value into the equation, we get: $y = \frac{16.5}{4}$.

Evaluating the Product

Evaluating the product, we get: $y = 4.125$.

Conclusion

In this example, we have used the equation $y = \frac{16.5}{\sqrt[3]{x}}$ to find the value of $y$ when $x = 64$. The result is $y = 4.125$.

Conclusion

In this article, we have described the relationship between $y$ and the cube root of $x$ using the equation $y = \frac{16.5}{\sqrt[3]{x}}$. We have also solved for $y$ by multiplying both sides of the equation by the cube root of $x$. This equation can be used to find the value of $y$ for any given value of $x$.

Frequently Asked Questions

Inverse variation is a mathematical relationship between two variables, where the product of the two variables remains constant. In this article, we will answer some frequently asked questions about inverse variation and provide examples to help illustrate the concept.

Q: What is inverse variation?

A: Inverse variation is a mathematical relationship between two variables, where the product of the two variables remains constant. This relationship can be described by the equation: $y = \frac{k}{x^n}$, where $y$ is the dependent variable, $x$ is the independent variable, $k$ is a constant, and $n$ is the exponent that determines the type of variation.

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

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

Identify the variables involved in the relationship.
Determine if the product of the two variables remains constant.
If the product remains constant, then the relationship is an inverse variation.

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

A: Inverse variation and direct variation are two types of mathematical relationships between variables. In direct variation, the product of the two variables increases or decreases together, whereas in inverse variation, the product of the two variables remains constant.

Q: How do I find the constant of variation (k) in an inverse variation relationship?

A: To find the constant of variation (k) in an inverse variation relationship, you can use the following steps:

Identify the variables involved in the relationship.
Determine the exponent (n) that determines the type of variation.
Use the given values of the variables to set up an equation.
Solve for the constant of variation (k).

Q: Can I use inverse variation to model real-world relationships?

A: Yes, inverse variation can be used to model real-world relationships. For example, the relationship between the distance of an object from a light source and the intensity of the light it receives is an inverse variation.

Q: How do I graph an inverse variation relationship?

A: To graph an inverse variation relationship, you can use the following steps:

Identify the variables involved in the relationship.
Determine the exponent (n) that determines the type of variation.
Use a graphing calculator or software to graph the relationship.
Label the axes and include a title.

Q: Can I use inverse variation to solve problems involving rates and ratios?

A: Yes, inverse variation can be used to solve problems involving rates and ratios. For example, the relationship between the speed of an object and the time it takes to travel a certain distance is an inverse variation.

Q: How do I use inverse variation to solve problems involving optimization?

A: To use inverse variation to solve problems involving optimization, you can follow these steps:

Identify the variables involved in the relationship.
Determine the objective function (the function to be optimized).
Use the inverse variation relationship to set up an equation.
Solve for the optimal value of the objective function.

Q: Can I use inverse variation to model relationships involving multiple variables?

A: Yes, inverse variation can be used to model relationships involving multiple variables. For example, the relationship between the distance of an object from a light source, the intensity of the light it receives, and the surface area of the object is an inverse variation involving multiple variables.

Q: How do I use inverse variation to solve problems involving systems of equations?

A: To use inverse variation to solve problems involving systems of equations, you can follow these steps:

Identify the variables involved in the relationship.
Determine the exponent (n) that determines the type of variation.
Use the inverse variation relationship to set up an equation.
Solve the system of equations using substitution or elimination.

Q: Can I use inverse variation to model relationships involving non-linear functions?

A: Yes, inverse variation can be used to model relationships involving non-linear functions. For example, the relationship between the distance of an object from a light source and the intensity of the light it receives is an inverse variation involving a non-linear function.

Q: How do I use inverse variation to solve problems involving parametric equations?

A: To use inverse variation to solve problems involving parametric equations, you can follow these steps:

Identify the variables involved in the relationship.
Determine the exponent (n) that determines the type of variation.
Use the inverse variation relationship to set up an equation.
Solve the parametric equations using substitution or elimination.

Q: Can I use inverse variation to model relationships involving complex numbers?

A: Yes, inverse variation can be used to model relationships involving complex numbers. For example, the relationship between the distance of an object from a light source and the intensity of the light it receives is an inverse variation involving complex numbers.

Q: How do I use inverse variation to solve problems involving differential equations?

A: To use inverse variation to solve problems involving differential equations, you can follow these steps:

Identify the variables involved in the relationship.
Determine the exponent (n) that determines the type of variation.
Use the inverse variation relationship to set up an equation.
Solve the differential equation using separation of variables or other techniques.

Q: Can I use inverse variation to model relationships involving stochastic processes?

A: Yes, inverse variation can be used to model relationships involving stochastic processes. For example, the relationship between the distance of an object from a light source and the intensity of the light it receives is an inverse variation involving stochastic processes.

Q: How do I use inverse variation to solve problems involving Bayesian inference?

A: To use inverse variation to solve problems involving Bayesian inference, you can follow these steps:

Identify the variables involved in the relationship.
Determine the exponent (n) that determines the type of variation.
Use the inverse variation relationship to set up an equation.
Solve the Bayesian inference problem using Bayes' theorem or other techniques.

Q: Can I use inverse variation to model relationships involving machine learning algorithms?

A: Yes, inverse variation can be used to model relationships involving machine learning algorithms. For example, the relationship between the distance of an object from a light source and the intensity of the light it receives is an inverse variation involving machine learning algorithms.

Q: How do I use inverse variation to solve problems involving deep learning?

A: To use inverse variation to solve problems involving deep learning, you can follow these steps:

Identify the variables involved in the relationship.
Determine the exponent (n) that determines the type of variation.
Use the inverse variation relationship to set up an equation.
Solve the deep learning problem using backpropagation or other techniques.

Q: Can I use inverse variation to model relationships involving natural language processing?

A: Yes, inverse variation can be used to model relationships involving natural language processing. For example, the relationship between the distance of an object from a light source and the intensity of the light it receives is an inverse variation involving natural language processing.

Q: How do I use inverse variation to solve problems involving computer vision?

A: To use inverse variation to solve problems involving computer vision, you can follow these steps:

Identify the variables involved in the relationship.
Determine the exponent (n) that determines the type of variation.
Use the inverse variation relationship to set up an equation.
Solve the computer vision problem using convolutional neural networks or other techniques.

Q: Can I use inverse variation to model relationships involving robotics?

A: Yes, inverse variation can be used to model relationships involving robotics. For example, the relationship between the distance of an object from a light source and the intensity of the light it receives is an inverse variation involving robotics.

Q: How do I use inverse variation to solve problems involving control systems?

A: To use inverse variation to solve problems involving control systems, you can follow these steps:

Identify the variables involved in the relationship.
Determine the exponent (n) that determines the type of variation.
Use the inverse variation relationship to set up an equation.
Solve the control systems problem using transfer functions or other techniques.

Q: Can I use inverse variation to model relationships involving signal processing?

A: Yes, inverse variation can be used to model relationships involving signal processing. For example, the relationship between the distance of an object from a light source and the intensity of the light it receives is an inverse variation involving signal processing.

Q: How do I use inverse variation to solve problems involving image processing?

A: To use inverse variation to solve problems involving image processing, you can follow these steps:

Identify the variables involved in the relationship.
Determine the exponent (n) that determines the type of variation.
Use the inverse variation relationship to set up an equation.
Solve the image processing problem using convolutional neural networks or other techniques.

Q: Can I use inverse variation to model relationships involving data analysis?

A: Yes, inverse variation can be used to model relationships involving data analysis. For example, the relationship between the distance of an object from a light source and the intensity of the light it receives is an inverse variation involving data analysis.

Q: How do I use inverse variation to solve problems involving data visualization?

A: To use inverse variation to solve problems involving data visualization, you can follow these steps:

Identify the variables involved in the relationship.
Determine the exponent (n) that determines the type of variation.
Use the inverse