Find The Constant Of Variation $k$ If $c$ Varies Directly With $a$ And Inversely With $b$, Given That $c=\frac{3}{20}$ When $a=2$ And $b=5$.

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In the fascinating world of mathematics, direct and inverse variation play a crucial role in describing relationships between variables. Understanding these concepts is essential for solving a wide range of problems in various fields, from physics and engineering to economics and everyday life. In this article, we will delve into the concept of direct and inverse variation, and explore how to determine the constant of variation in a given scenario.

Direct Variation: A Proportional Relationship

Direct variation describes a relationship where one variable increases proportionally with another. In other words, as one variable increases, the other variable increases at a constant rate. This relationship can be expressed mathematically as follows:

y=kxy = kx

where:

  • y is the dependent variable
  • x is the independent variable
  • k is the constant of variation

The constant of variation, k, represents the factor by which y changes for every unit change in x. It is a crucial value that defines the specific direct variation relationship between the two variables. To find the constant of variation, you need to know at least one pair of corresponding values for x and y. Once you have these values, you can simply substitute them into the equation and solve for k.

For example, let's say the distance traveled by a car varies directly with the time it travels. If the car travels 100 miles in 2 hours, we can find the constant of variation as follows:

100=k∗2100 = k * 2

k=50k = 50

This means that the car travels 50 miles per hour, which is the constant of variation in this case.

Inverse Variation: An Inversely Proportional Relationship

Inverse variation, on the other hand, describes a relationship where one variable decreases as the other variable increases, and vice versa. This relationship can be expressed mathematically as follows:

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

where:

  • y is the dependent variable
  • x is the independent variable
  • k is the constant of variation

In inverse variation, the constant of variation, k, represents the product of x and y. Similar to direct variation, you need to know at least one pair of corresponding values for x and y to determine the constant of variation. Once you have these values, you can substitute them into the equation and solve for k.

For instance, consider the relationship between the number of workers and the time it takes to complete a task. If it takes 4 workers 6 hours to complete a task, we can find the constant of variation as follows:

6=k46 = \frac{k}{4}

k=24k = 24

This indicates that the task requires 24 worker-hours, which is the constant of variation in this scenario.

Combining Direct and Inverse Variation

In many real-world situations, variables can be related through a combination of both direct and inverse variation. This means that one variable might vary directly with one variable and inversely with another. Such relationships can be expressed mathematically as follows:

z=kabz = \frac{ka}{b}

where:

  • z is the dependent variable
  • a and b are independent variables
  • k is the constant of variation

In this equation, z varies directly with a and inversely with b. To determine the constant of variation, you need to know at least one set of corresponding values for z, a, and b. Once you have these values, you can substitute them into the equation and solve for k.

Solving for the Constant of Variation: A Step-by-Step Approach

Let's consider a problem where we are given that the variable $c$ varies directly with $a$ and inversely with $b$, and $c=\frac{3}{20}$ when $a=2$ and $b=5$. Our goal is to find the constant of variation, k.

To solve this problem, we will follow these steps:

  1. Write the general equation:

    Since $c$ varies directly with $a$ and inversely with $b$, we can write the equation as:

    c=kabc = \frac{ka}{b}

    This equation represents the combined direct and inverse variation relationship between the variables.

  2. Substitute the given values:

    We are given that $c=\frac{3}{20}$, $a=2$, and $b=5$. Substituting these values into the equation, we get:

    320=k∗25\frac{3}{20} = \frac{k * 2}{5}

    This step replaces the variables with their known values, allowing us to isolate and solve for the constant of variation.

  3. Solve for k:

    To solve for k, we can multiply both sides of the equation by $\frac{5}{2}$:

    320∗52=k\frac{3}{20} * \frac{5}{2} = k

    Simplifying the equation, we get:

    k=38k = \frac{3}{8}

    Therefore, the constant of variation in this case is $\frac{3}{8}$.

Significance of the Constant of Variation

The constant of variation, k, plays a crucial role in understanding and interpreting direct and inverse variation relationships. It provides a specific numerical value that quantifies the relationship between the variables. In direct variation, k represents the constant rate of change between the variables, while in inverse variation, k represents the constant product of the variables.

Knowing the constant of variation allows us to:

  • Predict the value of one variable given the value of another: Once we know the constant of variation, we can easily calculate the value of the dependent variable for any given value of the independent variable, and vice versa.
  • Compare different relationships: By comparing the constants of variation for different relationships, we can determine which relationship is stronger or weaker.
  • Model real-world phenomena: Direct and inverse variation relationships are commonly used to model real-world phenomena in various fields, and the constant of variation provides a key parameter for these models.

Conclusion

Direct and inverse variation are fundamental concepts in mathematics that describe proportional relationships between variables. Understanding these concepts and how to determine the constant of variation is essential for solving a wide range of problems. By following a step-by-step approach, we can easily find the constant of variation in any given scenario. The constant of variation provides valuable insights into the relationship between variables and allows us to make predictions and comparisons.

Keywords: direct variation, inverse variation, constant of variation, proportional relationship, inversely proportional relationship, mathematical modeling, problem-solving.