Problem 38 - A Graphical Approach to College Algebra

Question

In Exercises \(37-40,\) find the constant of variation \(k\) and the undetermined value in the table if \(y\) is directly proportional to \(x\). $$\begin{array}{c|c|c|c|c} x & 1.2 & 4.3 & 5.7 & ? \\ \hline y & 3.96 & 14.19 & 18.81 & 23.43 \end{array}$$

Step-by-Step Solution

Answer

The constant of variation \(k = 3.3\). The missing value of \(x\) is 7.1.

Step 1: Understanding Direct Proportionality

When two quantities, say \(y\) and \(x\), are directly proportional, their ratio \(\frac{y}{x}\) is a constant. This constant is known as the constant of variation, \(k\). So if \(y = kx\), then \(k = \frac{y}{x}\).

Step 2: Calculate Constant of Variation \(k\) Using Known Values

We are given values \(x = 1.2\) and \(y = 3.96\). Using these values, calculate \(k\) using the formula \(k = \frac{y}{x}\). Substitute the values: \[ k = \frac{3.96}{1.2} = 3.3. \]

Step 3: Verify Constant \(k\) with Additional Data Points

Verifying with \(x = 4.3\) and \(y = 14.19\), calculate \(k\):\[ k = \frac{14.19}{4.3} \approx 3.3. \] The constant \(k\) remains consistent with the other values, suggesting no calculation errors.

Step 4: Apply \(k\) to Determine the Missing \(x\) Value

Given \(y = 23.43\), and \(k = 3.3\), find \(x\) using \(y = kx\): Rearrange the equation and solve for \(x\): \[ x = \frac{y}{k} = \frac{23.43}{3.3} \approx 7.1. \]

Step 5: Confirm the Solution

Cross-verify \(x = 7.1\) with the derived equation \(y = kx\) to ensure correctness. Since \(23.43 \approx 3.3 \times 7.1\), the calculation matches with the given \(y\) value, confirming the correctness of \(x = 7.1\).

Key Concepts

Constant of Variation

In direct proportionality, the constant of variation, often denoted by the symbol \( k \), plays a fundamental role. When two variables, say \( y \) and \( x \), are directly proportional, it means that as one quantity increases, the other does so at a consistent rate. This relationship can be captured through the equation \( y = kx \), where \( k \) is the constant of variation.

The constant of variation tells us how much \( y \) changes for a unit change in \( x \). For instance, if \( k = 3.3 \) as derived from our data, it implies that for every single unit increase in \( x \), \( y \) increases by 3.3 units.

To find \( k \), you divide the value of \( y \) by \( x \), giving us the formula \( k = \frac{y}{x} \). This constant should remain the same for all given pairs \( (x, y) \) in a perfectly directly proportional relationship, indicating consistency and correctness in our calculations.

Ratio

A crucial component of directly proportional relationships is the ratio of the variables involved. The ratio is the division of one quantity by another, and in direct proportions, this ratio remains a constant value across different sets of data.

For example, if we have data points \( (1.2, 3.96) \) and \( (4.3, 14.19) \), and the ratio \( \frac{y}{x} \) for both cases yields \( 3.3 \), we verify that these two quantities are directly proportional with the constant \( k = 3.3 \). This consistency showcases that the change in \( y \) compared to the change in \( x \) is steady and predictable.

Understanding the ratio helps in verifying the direct proportionality and calculating missing values in a dataset with more confidence.

Algebraic Equation

Algebraic equations form the backbone of solving direct proportionality problems. In our scenario, the equation \( y = k x \) seamlessly connects the two variables. By using this simple yet powerful formula, we determine unknowns such as \( k \) or an undetermined variable.

Take, for instance, the problem of finding a missing \( x \) value when \( y = 23.43 \). Knowing \( k = 3.3 \), we rearrange the algebraic equation to \( x = \frac{y}{k} \), then substitute the known values:

\[ x = \frac{23.43}{3.3} \]
\[ \approx 7.1 \]
This straightforward application of algebraic manipulation enables us to solve for unknowns rapidly and efficiently. Such equations provide a structured approach to solving many types of problems where relationships between variables need to be explored.