Deriving the Variation Equation

Oct 13, 2025 by windailys.com 52 views

Introduction

In mathematics, inverse variation is a fundamental concept 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, focusing on the relationship between two variables, $S$ and $G$ . We will derive the variation equation, find the value of $S$ when $G$ is 3, and discuss the nature of this relationship.

What is Inverse Variation?

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

S \propto \frac{1}{G}

where $S$ and $G$ are the variables, and $k$ is the constant of proportionality.

Deriving the Variation Equation

Given that $S$ varies inversely as $G$ , we can write the variation equation as:

S = \frac{k}{G}

To find the value of $k$ , we can use the given information that $S$ is 6 when $G$ is 0.6. Substituting these values into the equation, we get:

6 = \frac{k}{0.6}

Solving for $k$ , we get:

k = 6 \times 0.6 = 3.6

Therefore, the variation equation is:

S = \frac{3.6}{G}

Finding $S$ when $G$ is 3

To find the value of $S$ when $G$ is 3, we can substitute $G = 3$ into the variation equation:

S = \frac{3.6}{3} = 1.2

Therefore, when $G$ is 3, the value of $S$ is 1.2.

How are these two variables related?

The relationship between $S$ and $G$ is an inverse variation relationship. As $G$ increases, $S$ decreases, and vice versa. This means that if $G$ doubles, $S$ will halve, and if $G$ triples, $S$ will become one-third of its original value.

Graphical Representation

The inverse variation relationship between $S$ and $G$ can be represented graphically as a hyperbola. The graph will have a characteristic "S" shape, with the asymptotes intersecting at the origin.

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.
Economics: Inverse variation is used to describe the relationship between the price of a commodity and its demand.
Biology: Inverse variation is used to describe the relationship between the concentration of a substance and its rate of diffusion.

Conclusion

In conclusion, inverse variation is a fundamental concept in mathematics that describes the relationship between two variables. We have derived the variation equation, found the value of $S$ when $G$ is 3, and discussed the nature of this relationship. Inverse variation has numerous real-world applications, and understanding this concept is essential for problem-solving in various fields.

Frequently Asked Questions

Q: What is inverse variation?

A: Inverse variation is a type of functional relationship where the product of the two variables remains constant.

Q: How do you derive the variation equation?

A: To derive the variation equation, you need to use the given information and substitute the values into the equation.

Q: What is the relationship between $S$ and $G$ ?

A: The relationship between $S$ and $G$ is an inverse variation relationship, where as $G$ increases, $S$ decreases, and vice versa.

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

A: Inverse variation has numerous real-world applications, including physics, economics, and biology.

Q: How do you represent inverse variation graphically?

A: The inverse variation relationship between $S$ and $G$ can be represented graphically as a hyperbola.