The Square of p Varies Inversely as q

Oct 13, 2025 by windailys.com 15 views

Understanding Inverse Variation

Inverse variation is a fundamental concept in mathematics that describes the relationship between two variables. It states that as one variable increases, the other variable decreases, and vice versa. In this article, we will explore the concept of inverse variation and its applications, particularly in the context of the given problem.

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 $y$ and $x$ are the variables, and $k$ is a constant.

The Square of p Varies Inversely as q

In the given problem, we are told that the square of $p$ varies inversely as $q$ . This can be represented as:

p^2 = \frac{k}{q}

where $k$ is a constant.

Given Values

We are given that when $q=6$ , $p=4$ . We can use this information to find the value of $k$ .

p^2 = \frac{k}{q}

4^2 = \frac{k}{6}

16 = \frac{k}{6}

k = 96

Finding the Value of q

Now that we have found the value of $k$ , we can use it to find the value of $q$ when $p=12$ .

p^2 = \frac{k}{q}

12^2 = \frac{96}{q}

144 = \frac{96}{q}

q = \frac{96}{144}

q = \frac{2}{3}

Conclusion

In this article, we have explored the concept of inverse variation and its applications. We have used the given problem to find the value of $q$ when $p=12$ . The correct answer is $\frac{2}{3}$ .

Discussion

The concept of inverse variation is widely used in various fields, including physics, engineering, and economics. It helps us understand the relationships between different variables and make predictions about their behavior.

Common Mistakes

When working with inverse variation, it's essential to remember that the product of the two variables remains constant. This means that if one variable increases, the other variable must decrease, and vice versa.

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 specific relationship between variables, such as the relationship between the speed of a car and the distance traveled.
Economics: Inverse variation is used to model the relationship between the price of a good and the quantity demanded.

Conclusion

In conclusion, inverse variation is a fundamental concept in mathematics that describes the relationship between two variables. It has numerous real-world applications and is widely used in various fields. By understanding inverse variation, we can make predictions about the behavior of different variables and design systems that require a specific relationship between them.

Final Answer

Frequently Asked Questions

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 $y$ and $x$ are the variables, and $k$ is a constant.

Q: How do I identify inverse variation?

A: To identify inverse variation, look for a relationship where one variable increases and the other variable decreases, and vice versa. You can also use the following characteristics:

The product of the two variables remains constant.
The graph of the relationship is a hyperbola.
The equation of the relationship is of the form $y = \frac{k}{x}$ .

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 specific relationship between variables, such as the relationship between the speed of a car and the distance traveled.
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, follow these steps:

Write the equation of the relationship in the form $y = \frac{k}{x}$ .
Use the given information to find the value of $k$ .
Use the value of $k$ to find the value of the other variable.

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:

Confusing inverse variation with direct variation: Make sure to check the relationship between the variables before assuming it's inverse variation.
Not checking for a constant product: Make sure the product of the two variables remains constant.
Not using the correct equation: Use the equation $y = \frac{k}{x}$ to represent inverse variation.

Q: How do I graph inverse variation?

A: To graph inverse variation, follow these steps:

Plot the points on a coordinate plane.
Draw a hyperbola that passes through the points.
Label the axes and the equation of the relationship.

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

A: Some tips for working with inverse variation include:

Use a calculator to check your work: Use a calculator to check your work and ensure that the product of the two variables remains constant.
Check for a constant product: Make sure the product of the two variables remains constant.
Use the correct equation: Use the equation $y = \frac{k}{x}$ to represent inverse variation.

Conclusion

Final Answer

The final answer is $\boxed{\frac{2}{3}}$ .