Suppose That $x$ And $y$ Vary Inversely. When $x = 3$, $y = 12$.a) Find The Inverse Constant $k$.$k = \square$b) Find The $x$ Value When $y = 9$.$x = \square$

Understanding 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:

$$y = \frac{k}{x}$$

where $y$ is the dependent variable, $x$ is the independent variable, and $k$ is the inverse constant.

Finding the Inverse Constant $k$

Given that $x$ and $y$ vary inversely, and when $x = 3$, $y = 12$, we can use this information to find the value of the inverse constant $k$.

To find $k$, we can substitute the given values of $x$ and $y$ into the equation:

$$y = \frac{k}{x}$$

$$12 = \frac{k}{3}$$

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

$$k = 12 \times 3$$

$$k = 36$$

Therefore, the value of the inverse constant $k$ is $36$.

Finding the $x$ Value When $y = 9$

Now that we have found the value of $k$, we can use it to find the $x$ value when $y = 9$.

We can substitute the value of $k$ and the given value of $y$ into the equation:

$$y = \frac{k}{x}$$

$$9 = \frac{36}{x}$$

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

$$9x = 36$$

Next, we can divide both sides of the equation by $9$:

$$x = \frac{36}{9}$$

$$x = 4$$

Therefore, the value of $x$ when $y = 9$ is $4$.

Real-World Applications of Inverse Variation

Inverse variation has many real-world applications, including:

• Physics: The force of gravity between two objects varies inversely with the square of the distance between them.
• Economics: The demand for a product may vary inversely with its price.
• Biology: The rate of photosynthesis may vary inversely with the concentration of carbon dioxide in the air.

Conclusion

In this article, we have discussed the concept of inverse variation and how to find the inverse constant $k$ and solve for unknown values. We have also explored some real-world applications of inverse variation. By understanding inverse variation, we can better analyze and model complex relationships in various fields.

Example Problems

Problem 1

Suppose that $x$ and $y$ vary inversely. When $x = 2$, $y = 8$. Find the inverse constant $k$.

Solution

To find $k$, we can substitute the given values of $x$ and $y$ into the equation:

$$y = \frac{k}{x}$$

$$8 = \frac{k}{2}$$

To solve for $k$, we can multiply both sides of the equation by $2$:

$$k = 8 \times 2$$

$$k = 16$$

Therefore, the value of the inverse constant $k$ is $16$.

Problem 2

Suppose that $x$ and $y$ vary inversely. When $y = 6$, $x = 2$. Find the inverse constant $k$.

Solution

To find $k$, we can substitute the given values of $x$ and $y$ into the equation:

$$y = \frac{k}{x}$$

$$6 = \frac{k}{2}$$

To solve for $k$, we can multiply both sides of the equation by $2$:

$$k = 6 \times 2$$

$$k = 12$$

Therefore, the value of the inverse constant $k$ is $12$.

Practice Problems

Problem 1

Suppose that $x$ and $y$ vary inversely. When $x = 4$, $y = 12$. Find the inverse constant $k$.

Problem 2

Suppose that $x$ and $y$ vary inversely. When $y = 9$, $x = 3$. Find the inverse constant $k$.

Problem 3

Suppose that $x$ and $y$ vary inversely. When $x = 2$, $y = 4$. Find the inverse constant $k$.

Problem 4

Suppose that $x$ and $y$ vary inversely. When $y = 6$, $x = 3$. Find the inverse constant $k$.

Solutions

Problem 1

To find $k$, we can substitute the given values of $x$ and $y$ into the equation:

$$y = \frac{k}{x}$$

$$12 = \frac{k}{4}$$

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

$$k = 12 \times 4$$

$$k = 48$$

Therefore, the value of the inverse constant $k$ is $48$.

Problem 2

To find $k$, we can substitute the given values of $x$ and $y$ into the equation:

$$y = \frac{k}{x}$$

$$9 = \frac{k}{3}$$

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

$$k = 9 \times 3$$

$$k = 27$$

Therefore, the value of the inverse constant $k$ is $27$.

Problem 3

To find $k$, we can substitute the given values of $x$ and $y$ into the equation:

$$y = \frac{k}{x}$$

$$4 = \frac{k}{2}$$

To solve for $k$, we can multiply both sides of the equation by $2$:

$$k = 4 \times 2$$

$$k = 8$$

Therefore, the value of the inverse constant $k$ is $8$.

Problem 4

To find $k$, we can substitute the given values of $x$ and $y$ into the equation:

$$y = \frac{k}{x}$$

$$6 = \frac{k}{3}$$

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

$$k = 6 \times 3$$

$$k = 18$$

$$$$

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:

$$y = \frac{k}{x}$$

where $y$ is the dependent variable, $x$ is the independent variable, and $k$ is the inverse constant.

Q: How do I find the inverse constant $k$?

A: To find the inverse constant $k$, you can substitute the given values of $x$ and $y$ into the equation:

$$y = \frac{k}{x}$$

Then, you can solve for $k$ by multiplying both sides of the equation by $x$.

Q: What are some real-world applications of inverse variation?

A: Inverse variation has many real-world applications, including:

• Physics: The force of gravity between two objects varies inversely with the square of the distance between them.
• Economics: The demand for a product may vary inversely with its price.
• Biology: The rate of photosynthesis may vary inversely with the concentration of carbon dioxide in the air.

Q: How do I solve for $x$ when $y$ is given?

A: To solve for $x$ when $y$ is given, you can substitute the value of $y$ and the inverse constant $k$ into the equation:

$$y = \frac{k}{x}$$

Then, you can solve for $x$ by multiplying both sides of the equation by $x$ and then dividing both sides by $y$.

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

A: Some common mistakes to avoid when working with inverse variation include:

• Not checking units: Make sure that the units of the variables are consistent.
• Not checking for extraneous solutions: Make sure that the solutions you find are not extraneous.
• Not using the correct equation: Make sure that you are using the correct equation for inverse variation.

Q: How do I graph an inverse variation?

A: To graph an inverse variation, you can use a graphing calculator or a graphing software. You can also use a table of values to create a graph.

Q: What are some tips for working with inverse variation?

A: Some tips for working with inverse variation include:

• Read the problem carefully: Make sure that you understand the problem and what is being asked.
• Check your work: Make sure that your solutions are correct and that you have not made any mistakes.
• Use a graphing calculator or software: A graphing calculator or software can be a big help when working with inverse variation.

Q: How do I use inverse variation to model real-world problems?

A: To use inverse variation to model real-world problems, you can:

• Identify the variables: Identify the variables in the problem and determine which one is the independent variable and which one is the dependent variable.
• Determine the equation: Determine the equation of the inverse variation based on the problem.
• Solve for the unknown: Solve for the unknown variable using the equation.

Q: What are some common applications of inverse variation in science and engineering?

A: Some common applications of inverse variation in science and engineering include:

• Physics: The force of gravity between two objects varies inversely with the square of the distance between them.
• Economics: The demand for a product may vary inversely with its price.
• Biology: The rate of photosynthesis may vary inversely with the concentration of carbon dioxide in the air.

Q: How do I use inverse variation to solve problems in finance?

A: To use inverse variation to solve problems in finance, you can:

• Identify the variables: Identify the variables in the problem and determine which one is the independent variable and which one is the dependent variable.
• Determine the equation: Determine the equation of the inverse variation based on the problem.
• Solve for the unknown: Solve for the unknown variable using the equation.

Q: What are some common applications of inverse variation in business?

A: Some common applications of inverse variation in business include:

• Marketing: The demand for a product may vary inversely with its price.
• Supply chain management: The cost of shipping a product may vary inversely with the distance between the supplier and the customer.
• Operations management: The time it takes to complete a task may vary inversely with the number of workers available.