The Quantity { A$}$ Varies Directly With { C$}$ And Inversely With { B$}$. When { C = -4$}$ And { B = -3$}$, { A = 16$}$. What Is The Equation Of Variation?Write Your Answer In The Form
Introduction
In mathematics, the concept of variation is a fundamental idea that helps us understand how different quantities are related to each other. In this article, we will explore the equation of variation, which describes the relationship between two or more variables. Specifically, we will examine a scenario where the quantity {a$}$ varies directly with {c$}$ and inversely with {b$}$. Our goal is to determine the equation of variation that describes this relationship.
Direct and Inverse Variation
Before we dive into the specifics of this problem, let's review the concepts of direct and inverse variation.
- Direct Variation: When two quantities are directly related, an increase in one quantity results in a proportional increase in the other quantity. Mathematically, this can be represented as {a = k \cdot c$}$, where {k$}$ is a constant of proportionality.
- Inverse Variation: When two quantities are inversely related, an increase in one quantity results in a proportional decrease in the other quantity. Mathematically, this can be represented as {a = \frac{k}{c}$}$, where {k$}$ is a constant of proportionality.
The Problem
We are given that the quantity {a$}$ varies directly with {c$}$ and inversely with {b$}$. In other words, we can represent this relationship as {a = k \cdot \frac{c}{b}$}$. When {c = -4$}$ and {b = -3$}$, we know that {a = 16$}$. Our goal is to determine the equation of variation that describes this relationship.
Solving for the Constant of Proportionality
To find the equation of variation, we need to determine the constant of proportionality {k$}$. We can do this by substituting the given values into the equation and solving for {k$}$.
${16 = k \cdot \frac{-4}{-3}\$}
Simplifying the equation, we get:
${16 = k \cdot \frac{4}{3}\$}
Multiplying both sides by {\frac{3}{4}$}$, we get:
{k = 16 \cdot \frac{3}{4}$}$
{k = 12$}$
The Equation of Variation
Now that we have found the constant of proportionality {k$}$, we can write the equation of variation that describes the relationship between {a$}$, {c$}$, and {b$}$.
{a = 12 \cdot \frac{c}{b}$}$
This equation tells us that the quantity {a$}$ varies directly with {c$}$ and inversely with {b$}$. In other words, as {c$}$ increases, {a$}$ also increases, but as {b$}$ increases, {a$}$ decreases.
Conclusion
In this article, we explored the concept of variation and determined the equation of variation that describes the relationship between {a$}$, {c$}$, and {b$}$. We found that the quantity {a$}$ varies directly with {c$}$ and inversely with {b$}$, and we wrote the equation of variation as {a = 12 \cdot \frac{c}{b}$}$. This equation provides a mathematical model for understanding the relationship between these three quantities.
Applications of the Equation of Variation
The equation of variation has numerous applications in various fields, including physics, engineering, economics, and more. For example, in physics, the equation of variation can be used to describe the relationship between the force of gravity and the distance between two objects. In engineering, the equation of variation can be used to design systems that respond to changes in input variables.
Real-World Examples
Here are a few real-world examples of the equation of variation in action:
- Sound Waves: The intensity of a sound wave varies inversely with the distance from the source of the sound. Mathematically, this can be represented as {I = \frac{k}{r^2}$}$, where {I$}$ is the intensity of the sound wave, {k$}$ is a constant of proportionality, and {r$}$ is the distance from the source of the sound.
- Electric Current: The resistance of a conductor varies directly with the length of the conductor and inversely with the cross-sectional area of the conductor. Mathematically, this can be represented as {R = \frac{k \cdot L}{A}$}$, where {R$}$ is the resistance of the conductor, {k$}$ is a constant of proportionality, {L$}$ is the length of the conductor, and {A$}$ is the cross-sectional area of the conductor.
Conclusion
Q: What is the equation of variation?
A: The equation of variation is a mathematical model that describes the relationship between two or more variables. It is used to express how one quantity varies in response to changes in another quantity.
Q: What are the different types of variation?
A: There are two main types of variation:
- Direct Variation: When two quantities are directly related, an increase in one quantity results in a proportional increase in the other quantity. Mathematically, this can be represented as {a = k \cdot c$}$, where {k$}$ is a constant of proportionality.
- Inverse Variation: When two quantities are inversely related, an increase in one quantity results in a proportional decrease in the other quantity. Mathematically, this can be represented as {a = \frac{k}{c}$}$, where {k$}$ is a constant of proportionality.
Q: How do I determine the equation of variation?
A: To determine the equation of variation, you need to identify the type of variation and then use the given data to find the constant of proportionality. Once you have the constant of proportionality, you can write the equation of variation.
Q: What is the constant of proportionality?
A: The constant of proportionality is a value that is used to describe the relationship between the variables. It is a measure of how much one variable changes in response to a change in the other variable.
Q: How do I find the constant of proportionality?
A: To find the constant of proportionality, you need to use the given data and the equation of variation. You can do this by substituting the given values into the equation and solving for the constant of proportionality.
Q: What are some real-world examples of the equation of variation?
A: Here are a few real-world examples of the equation of variation:
- Sound Waves: The intensity of a sound wave varies inversely with the distance from the source of the sound. Mathematically, this can be represented as {I = \frac{k}{r^2}$}$, where {I$}$ is the intensity of the sound wave, {k$}$ is a constant of proportionality, and {r$}$ is the distance from the source of the sound.
- Electric Current: The resistance of a conductor varies directly with the length of the conductor and inversely with the cross-sectional area of the conductor. Mathematically, this can be represented as {R = \frac{k \cdot L}{A}$}$, where {R$}$ is the resistance of the conductor, {k$}$ is a constant of proportionality, {L$}$ is the length of the conductor, and {A$}$ is the cross-sectional area of the conductor.
Q: What are some common mistakes to avoid when working with the equation of variation?
A: Here are a few common mistakes to avoid when working with the equation of variation:
- Not identifying the type of variation: Make sure to identify whether the relationship is direct or inverse variation.
- Not finding the constant of proportionality: Make sure to find the constant of proportionality using the given data and the equation of variation.
- Not using the correct units: Make sure to use the correct units when working with the equation of variation.
Q: How do I apply the equation of variation in real-world situations?
A: To apply the equation of variation in real-world situations, you need to identify the type of variation and then use the given data to find the constant of proportionality. Once you have the constant of proportionality, you can write the equation of variation and use it to make predictions or solve problems.
Conclusion
In conclusion, the equation of variation is a powerful tool for understanding the relationships between different quantities. By applying the concepts of direct and inverse variation, we can develop mathematical models that describe the behavior of complex systems. By avoiding common mistakes and applying the equation of variation in real-world situations, we can use this tool to make predictions, solve problems, and gain a deeper understanding of the world around us.