Problem 19 - College Algebra

Question

Determine the constant of variation for each stated condition. \(A\) varies directly as \(B\) and inversely as \(C,\) and \(A=9\) when \(B=12\) and \(C=4\)

Step-by-Step Solution

Answer

The constant of variation \(k\) is 3.

Step 1: Write down the variation formula

The formula for direct and inverse variation is \(A = kB/C\). In this case, \(A\), \(B\) and \(C\) represent the given quantities in the problem while \(k\) is the constant of variation that we want to find.

Step 2: Substitute given values

Now, we can substitute the given values \(A=9\), \(B=12\) and \(C=4\) into the formula. This yields \(9 = k*12/4\).

Step 3: Solve for constant of variation

To solve for \(k\), we'll rearrange the equation by multiplying by 4/12 on both sides of the equation. This gives us \(k = 9*(4/12) = 3\). So, the constant of variation \(k\) is 3.

Key Concepts

Direct Variation

Direct variation is a relationship between two variables where one variable is a constant multiple of the other. If variable \(A\) varies directly as \(B\), then you can express this relationship with the equation \(A = kB\). The constant \(k\) is known as the constant of variation. This means that as \(B\) increases, \(A\) increases proportionally, and vice versa.
For example, if \(A\) is doubled, \(B\) will also double if they are directly proportional. This concept is often used to model situations where changes in one quantity lead to proportional changes in another.

Inverse Variation

Inverse variation describes a relationship where one variable increases while the other decreases. This is explained by the equation \(A = \frac{k}{C}\). When \(A\) varies inversely as \(C\), it implies that as \(C\) becomes larger, \(A\) becomes smaller, provided \(k\) remains constant.
An everyday example could be the time needed to complete a task that is inversely related to the number of people working on it. If you double the number of workers, you may complete the task in half the time.

Constant of Variation

The constant of variation is a key element in both direct and inverse variation. It describes how much one variable changes in relation to another. In the general equation \(A = \frac{kB}{C}\), \(k\) is the constant that ties together the direct and inverse relationships.
In our given problem, after substituting the known values into the variation formula \(A = \frac{kB}{C}\), we solved for \(k\) and found it to be 3. This means that in this specific scenario, the relationship between \(A\), \(B\), and \(C\) is set by a fixed multiplier of 3. Therefore, the constant of variation provides a clear understanding of how \(A\) is related to both \(B\) and \(C\).
Understanding the constant of variation helps clarify how altering variables impacts outcomes in proportional relationships.