Problem 31 - Algebra and Trigonometry with Analytic Geometry

Question

Examine the expression for the given set of data points of the form \((x, y)\). Find the constant of variation and a formula that describes how \(y\) varies with respect to \(x\). $$ \begin{aligned} y / x ; &\\{(0.6,0.72),(1.2,1.44),(4.2,5.04),(7.1,8.52)\\\ &(9.3,11.16)\\} \end{aligned} $$

Step-by-Step Solution

Answer

The constant of variation is 1.2, and the formula is \( y = 1.2x \).

Step 1: Understanding Direct Variation

Direct variation implies that as one variable increases, the other increases proportionally, i.e., \( y = kx \), where \( k \) is the constant of variation.}To find the constant \( k \), we need to check if the ratio \( \frac{y}{x} \) is the same for all data points.

Step 2: Calculate Ratios

Calculate \( \frac{y}{x} \) for each data point to find \( k \):1. For \((0.6, 0.72)\), \( \frac{0.72}{0.6} = 1.2 \)2. For \((1.2, 1.44)\), \( \frac{1.44}{1.2} = 1.2 \)3. For \((4.2, 5.04)\), \( \frac{5.04}{4.2} = 1.2 \)4. For \((7.1, 8.52)\), \( \frac{8.52}{7.1} = 1.2 \)5. For \((9.3, 11.16)\), \( \frac{11.16}{9.3} = 1.2 \)

Step 3: Identify the Constant of Variation

The ratio \( \frac{y}{x} = 1.2 \) is consistent across all data points, indicating \( k = 1.2 \). Therefore, \( y \) varies directly with \( x \) by the constant of variation \( k = 1.2 \).

Step 4: Formulate the Variation Equation

Using the constant of variation, express the relationship of \( y \) with respect to \( x \) as \( y = 1.2x \).

Step 5: Final Check

Verify your formula with one of the data points:For \((1.2, 1.44)\), substituting into the equation gives \( y = 1.2 \times 1.2 = 1.44 \), which aligns with the given data point.

Key Concepts

Constant of Variation

In direct variation, the constant of variation is crucial because it dictates how variables relate to each other. In our problem, the constant of variation, denoted as \(k\), is the ratio of \(y\) to \(x\) across all given data points. This constant determines the slope of the line that represents the relationship between \(y\) and \(x\).
This can be found by dividing \(y\) by \(x\) for each data point and ensuring the result is consistent. In this exercise, the constant of variation \(k\) was repeatedly found to be 1.2. Thus, \(k = 1.2\) for this set of data.

Variation Equation

The variation equation forms the foundation of direct variation problems. It provides a mathematical formula to describe the relationship between the variables. In the scenario you're examining, the variation equation follows the form \(y = kx\).
With \(k\), the constant of variation established as 1.2 in this case, the variation equation becomes \(y = 1.2x\). This equation signifies that \(y\) is directly proportional to \(x\) and scales by the factor of 1.2. It is crucial to verify this equation with the data points to ensure its validity.

Data Points Analysis

Analyzing data points is essential for validating a direct variation relationship. Start by calculating \(\frac{y}{x}\) for each data point and see if they all produce the same \(k\). This consistency is proof that the data follows a direct variation pattern.
For the data points provided in the exercise, such as \((0.6, 0.72)\), \((1.2, 1.44)\), \( \text{etc.} \), it was found that each point gave us \(k = 1.2\). This uniformity confirms the direct variation equation \(y = 1.2x\) accurately models the relationship between \(y\) and \(x\).
In solving these, always check your calculations to ensure there are no deviations, which would indicate a lack of direct variation.

Proportional Relationship

A proportional relationship in this context means that both variables increase or decrease together according to a constant factor. Here, \(y\) increases as \(x\) increases, maintaining a consistent ratio determined by the constant \(k\).
In simpler terms, as \(x\) multiplies by a certain number, \(y\) will also multiply by that number, given by the formula \(y = 1.2x\). This constant proportionality is what defines a linear and direct relationship, simplifying prediction and calculation of \(y\) for any given \(x\).
By understanding this proportional relationship, one can efficiently verify data sets or predict values without having to measure them constantly.