Modeling Direct And Inverse Variation When C Varies Directly With B And Inversely With A

Jul 11, 2025 by ADMIN 89 views

When $c$ Varies Directly with $b$ and Inversely with $a$: Finding the Equation

In mathematics, understanding the relationships between variables is crucial for solving problems and modeling real-world scenarios. One such relationship is direct and inverse variation. This article delves into a specific problem involving direct and inverse variation, providing a comprehensive explanation and solution. We will explore how to identify the correct equation that models the given situation, ensuring a clear understanding of the underlying concepts.

Understanding Direct and Inverse Variation

Before diving into the problem, let's clarify the concepts of direct and inverse variation. These concepts are fundamental to understanding how quantities relate to each other.

Direct Variation: When a variable y varies directly with a variable x, it means that y increases as x increases, and y decreases as x decreases. This relationship can be represented by the equation y = kx, where k is the constant of variation. The constant k represents the fixed ratio between y and x. For example, the distance traveled at a constant speed varies directly with time; the longer the time, the greater the distance traveled.

Inverse Variation: When a variable y varies inversely with a variable x, it means that y decreases as x increases, and y increases as x decreases. This relationship can be represented by the equation y = k/x, where k is the constant of variation. In this case, k represents the product of y and x. An example of inverse variation is the time it takes to complete a journey versus the speed; the higher the speed, the less time it takes.

Combined Variation: In many real-world scenarios, a variable may vary directly with one variable and inversely with another. This is known as combined variation. In our problem, we have a combined variation scenario where c varies directly with b and inversely with a. Understanding combined variation allows us to model more complex relationships, making it a vital tool in mathematical problem-solving.

Problem Statement: Modeling Combined Variation

The problem states: When a = 6 and b = 22, c = 33. Given that c varies directly with b and inversely with a, we need to determine which equation correctly models this situation. The provided options are:

$\frac{c}{ab} = \frac{1}{4}$
$c(\overrightarrow{ab}) = 4356$
$\frac{ca}{b} = 9$
$\frac{cb}{a} = 121$

To solve this problem, we need to translate the given variation relationship into an equation and then use the provided values to find the constant of variation. This will allow us to identify the correct equation from the options. Let's break down the process step by step to ensure clarity and accuracy.

Step-by-Step Solution

Express the Variation as an Equation: Since c varies directly with b and inversely with a, we can express this relationship as:

$c = k \frac{b}{a}$

Here, k is the constant of variation. This equation tells us that c is proportional to b and inversely proportional to a. The constant k will help us define the specific relationship between these variables.
Substitute the Given Values to Find k: We are given that a = 6, b = 22, and c = 33. Substitute these values into the equation:

$33 = k \frac{22}{6}$

Now, we need to solve for k. This will give us the specific constant that relates c, b, and a in this scenario. Finding k is crucial for determining the correct equation.
Solve for k: To solve for k, we can multiply both sides of the equation by ${\frac{6}{22}}$ :

$k = 33 \times \frac{6}{22}$

Simplify the expression:

$k = 33 \times \frac{3}{11}$

$k = 3 \times 3$

$k = 9$

So, the constant of variation, k, is 9. This means that the relationship between c, b, and a is specifically defined by this constant value.
Write the Specific Equation: Now that we have found k, we can write the specific equation that models the situation:

$c = 9 \frac{b}{a}$

This equation is the mathematical representation of the relationship between c, b, and a given the conditions of the problem. This form will help us compare it with the provided options.
Compare with the Given Options: We need to manipulate our equation to match one of the given options. Let's rearrange the equation:

Multiply both sides by a:

$ca = 9b$

Divide both sides by b:

$\frac{ca}{b} = 9$

Comparing this with the given options, we see that it matches option 3.

Analyzing the Options

Now, let's analyze why the other options are incorrect. This will reinforce our understanding of the problem and the correct solution.

$\frac{c}{ab} = \frac{1}{4}$ :

If this equation were correct, we would have:

$c = \frac{1}{4}ab$

Substituting the given values, we get:

$33 = \frac{1}{4}(6)(22)$

$33 = \frac{1}{4}(132)$

$33 = 33$

This seems correct at first glance, but it represents c varying directly with both a and b, which is not what the problem states. It doesn't reflect the inverse relationship with a.
$c(\overrightarrow{ab}) = 4356$ :

This option is unclear due to the vector notation ${(\overrightarrow{ab})}$ . If we interpret it as c(ab) = 4356, then:

$33(6)(22) = 4356$

$4356 = 4356$

This is arithmetically correct, but it does not represent the correct variation relationship. It's a mere verification of the given values rather than a model of the variation.
$\frac{ca}{b} = 9$ :

As we derived in our solution, this equation correctly models the situation where c varies directly with b and inversely with a. Substituting the values:

$\frac{33(6)}{22} = 9$

$\frac{198}{22} = 9$

$9 = 9$

This confirms that this equation holds true for the given values and represents the correct relationship.
$\frac{cb}{a} = 121$ :

Substituting the given values:

$\frac{33(22)}{6} = 121$

$\frac{726}{6} = 121$

$121 = 121$

This is also arithmetically correct, but it implies that c varies directly with a/b, which is the inverse of the given relationship. Therefore, it does not correctly model the situation.

Conclusion: The Correct Equation

After a detailed analysis, we have determined that the equation that correctly models the situation where c varies directly with b and inversely with a is:

$\frac{ca}{b} = 9$

This equation accurately represents the given relationship and satisfies the provided values. Understanding direct, inverse, and combined variation is essential for solving mathematical problems and modeling real-world phenomena. By breaking down the problem step by step and carefully analyzing each option, we have confidently arrived at the correct solution. This problem highlights the importance of translating variation relationships into mathematical equations and using given values to find the constant of variation.

In summary, solving problems involving variations requires a clear understanding of the definitions and the ability to manipulate equations effectively. This article provided a comprehensive guide to solving such problems, ensuring that you are well-equipped to tackle similar challenges in the future. Remember to always translate the variation statement into an equation, find the constant of variation, and then verify your solution with the given conditions.