If $y$ Varies Inversely As $x$, Which Statement Must Be True?A. If $x$ Is $\frac{1}{2}$, Then $y$ Is 2.B. If $x$ Is $\frac{1}{2}$, Then $y$ Is

Inverse variation is a fundamental concept in mathematics that describes the relationship between two variables. When one variable increases, the other decreases, and vice versa. In this article, we will explore the concept of inverse variation and determine which statement must be true if $y$ varies inversely as $x$.

What is Inverse Variation?

Inverse variation is a type of functional relationship where the product of two variables remains constant. Mathematically, it can be represented as:

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

where $k$ is a constant. This equation shows that as $x$ increases, $y$ decreases, and vice versa.

Properties of Inverse Variation

There are several properties of inverse variation that are essential to understand:

• Constant Product: The product of $x$ and $y$ is always constant. This means that if $x$ increases, $y$ decreases, and vice versa.
• Inverse Relationship: As $x$ increases, $y$ decreases, and vice versa.
• Zero Product: When $x$ is zero, $y$ is undefined, and vice versa.

Analyzing the Statements

Now that we have a good understanding of inverse variation, let's analyze the statements given in the problem.

Statement A

If $x$ is $\frac{1}{2}$, then $y$ is 2.

To determine if this statement is true, we need to substitute $x = \frac{1}{2}$ into the equation $y = \frac{k}{x}$.

$$y = \frac{k}{\frac{1}{2}}$$

$$y = 2k$$

Since we don't know the value of $k$, we cannot determine the value of $y$. Therefore, statement A is not necessarily true.

Statement B

If $x$ is $\frac{1}{2}$, then $y$ is

To determine if this statement is true, we need to substitute $x = \frac{1}{2}$ into the equation $y = \frac{k}{x}$.

$$y = \frac{k}{\frac{1}{2}}$$

$$y = 2k$$

Since we don't know the value of $k$, we cannot determine the value of $y$. However, we can say that $y$ is proportional to $k$, and $x$ is inversely proportional to $y$.

Conclusion

In conclusion, if $y$ varies inversely as $x$, then the statement that must be true is that $y$ is proportional to $k$, and $x$ is inversely proportional to $y$. We cannot determine the exact value of $y$ without knowing the value of $k$.

Real-World Applications

Inverse variation has numerous real-world applications, including:

• Physics: Inverse variation is used to describe the relationship between the force of gravity and the distance between two objects.
• Engineering: Inverse variation is used to design systems that require a constant product, such as electrical circuits and mechanical systems.
• Economics: Inverse variation is used to model the relationship between the price of a good and the quantity demanded.

Solved Examples

Here are some solved examples to illustrate the concept of inverse variation:

Example 1

A ball is thrown upwards from the ground with an initial velocity of 20 m/s. The height of the ball is inversely proportional to the time it takes to reach the ground. If the height of the ball is 10 m, how long does it take to reach the ground?

Let $h$ be the height of the ball and $t$ be the time it takes to reach the ground. Since the height is inversely proportional to the time, we can write:

$$h = \frac{k}{t}$$

Substituting $h = 10$ m, we get:

$$10 = \frac{k}{t}$$

$$t = \frac{k}{10}$$

Since we don't know the value of $k$, we cannot determine the exact value of $t$. However, we can say that $t$ is inversely proportional to $k$.

Example 2

A company produces a certain product that requires a constant amount of raw materials. The cost of the raw materials is inversely proportional to the quantity produced. If the cost of the raw materials is $100, how many units can be produced?

Let $c$ be the cost of the raw materials and $q$ be the quantity produced. Since the cost is inversely proportional to the quantity produced, we can write:

$$c = \frac{k}{q}$$

Substituting $c = 100$, we get:

$$100 = \frac{k}{q}$$

$$q = \frac{k}{100}$$

Since we don't know the value of $k$, we cannot determine the exact value of $q$. However, we can say that $q$ is inversely proportional to $k$.

Practice Problems

Here are some practice problems to test your understanding of inverse variation:

Problem 1

A car travels from city A to city B at an average speed of 60 km/h. The distance between the two cities is inversely proportional to the time it takes to travel. If the distance is 120 km, how long does it take to travel?

Problem 2

A company produces a certain product that requires a constant amount of raw materials. The cost of the raw materials is inversely proportional to the quantity produced. If the cost of the raw materials is $200, how many units can be produced?

Problem 3

A ball is thrown upwards from the ground with an initial velocity of 30 m/s. The height of the ball is inversely proportional to the time it takes to reach the ground. If the height of the ball is 15 m, how long does it take to reach the ground?

Answer Key

Here are the answers to the practice problems:

Problem 1

The time it takes to travel is 2 hours.

Problem 2

The number of units that can be produced is 10.

Problem 3

The time it takes to reach the ground is 3 seconds.

Conclusion

Inverse variation is a fundamental concept in mathematics that describes the relationship between two variables. When one variable increases, the other decreases, and vice versa. In this article, we will answer some frequently asked questions about inverse variation.

Q: What is inverse variation?

A: Inverse variation is a type of functional relationship where the product of two variables remains constant. Mathematically, it can be represented as:

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

where $k$ is a constant.

Q: What are the properties of inverse variation?

A: There are several properties of inverse variation that are essential to understand:

• Constant Product: The product of $x$ and $y$ is always constant. This means that if $x$ increases, $y$ decreases, and vice versa.
• Inverse Relationship: As $x$ increases, $y$ decreases, and vice versa.
• Zero Product: When $x$ is zero, $y$ is undefined, and vice versa.

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

A: To determine if a relationship is an inverse variation, you need to check if the product of the two variables is constant. If the product is constant, then the relationship is an inverse variation.

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

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

• Physics: Inverse variation is used to describe the relationship between the force of gravity and the distance between two objects.
• Engineering: Inverse variation is used to design systems that require a constant product, such as electrical circuits and mechanical systems.
• Economics: Inverse variation is used to model the relationship between the price of a good and the quantity demanded.

Q: How do I solve inverse variation problems?

A: To solve inverse variation problems, you need to follow these steps:

1. Write the equation: Write the equation of the inverse variation, which is $y = \frac{k}{x}$.
2. Substitute values: Substitute the given values into the equation.
3. Solve for the unknown: Solve for the unknown variable.
4. Check the answer: Check the answer to make sure it is reasonable.

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

A: Here are some common mistakes to avoid when working with inverse variation:

• Not checking the product: Not checking the product of the two variables to see if it is constant.
• Not using the correct equation: Not using the correct equation of the inverse variation.
• Not solving for the unknown: Not solving for the unknown variable.
• Not checking the answer: Not checking the answer to make sure it is reasonable.

Q: How do I graph an inverse variation?

A: To graph an inverse variation, you need to follow these steps:

1. Plot the points: Plot the points on a coordinate plane.
2. Draw the line: Draw the line that passes through the points.
3. Label the axes: Label the axes with the variables.
4. Label the equation: Label the equation of the inverse variation.

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

A: Here are some tips for working with inverse variation:

• Use the correct equation: Use the correct equation of the inverse variation.
• Check the product: Check the product of the two variables to see if it is constant.
• Solve for the unknown: Solve for the unknown variable.
• Check the answer: Check the answer to make sure it is reasonable.

Conclusion

In conclusion, inverse variation is a fundamental concept in mathematics that describes the relationship between two variables. When one variable increases, the other decreases, and vice versa. In this article, we have answered some frequently asked questions about inverse variation, including what it is, its properties, how to determine if a relationship is an inverse variation, and how to solve inverse variation problems. We have also provided some tips for working with inverse variation and some common mistakes to avoid.