Problem 39 - 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 \(x\). Sales tax \(y\) on a purchase of \(x\) dollars \begin{tabular}{c|c|c|c} \(\boldsymbol{x}\) & \(\$ 25\) & \(\$ 55\) & \(?\) \\ \hline \(\boldsymbol{y}\) & \(\$ 1.50\) & \(\mathfrak{S} 3.30\) & \(\mathfrak{S} 5.10\) \end{tabular}

Step-by-Step Solution

Answer

The constant of variation \( k \) is 0.06 and the missing value of \( x \) is 85.

Step 1: Understanding Direct Proportionality

When two quantities are directly proportional, it means that as one quantity changes, the other changes in a consistent way relative to the first. Mathematically, this is expressed as \( y = kx \), where \( k \) is the constant of variation.

Step 2: Finding the Constant of Variation

To find \( k \), we use the given values where \( y \) is provided for a corresponding \( x \). From the table, when \( x = 25 \), \( y = 1.50 \). So, we substitute into the equation: \( 1.50 = k \times 25 \), solving gives the constant \( k = \frac{1.50}{25} = 0.06 \).

Step 3: Verifying the Constant of Variation

To ensure \( k = 0.06 \) is correct, check with another pair of values. For \( x = 55 \), \( y = 3.30 \), substitute to get \( 3.30 = 0.06 \times 55 \). Simplifying, we find both sides equal, confirming \( k = 0.06 \) is consistent.

Step 4: Finding the Missing Value of x

Now, find the missing \( x \) when \( y = 5.10 \). Using \( y = kx \) with \( k = 0.06 \), substitute to get \( 5.10 = 0.06x \). Solving for \( x \), divide both sides by 0.06 to find \( x = \frac{5.10}{0.06} = 85 \).

Key Concepts

Constant of Variation

In the world of mathematics, the concept of direct proportionality is central. When two quantities are directly proportional, the relationship between them can be described using a constant. This constant is known as the "Constant of Variation," often denoted by the letter \( k \). In simpler terms, it's a specific number that defines how one variable changes with another.When we say \( y \) is directly proportional to \( x \), it means that as \( x \) changes, \( y \) does so as well, maintaining a consistent ratio. This consistent factor is our constant of variation. If \( y = kx \), then regardless of the values of \( x \) and \( y \), \( k \) remains the same within that set of conditions.Now, let's put this into a practical scenario. Imagine someone buying items and paying sales tax. The amount of sales tax they pay (\( y \)) is directly proportional to the amount they spend (\( x \)). If they know the sales tax on a \(25 purchase is \)1.50, they can use this information to determine the constant \( k \). They simply divide the sales tax by the purchase amount: \( 1.50/25 = 0.06 \). This means for every dollar spent, 6% is added as sales tax.

Finding Missing Values

When dealing with direct proportionality, you might encounter situations where certain values are missing, and your task is to find them. Once the "Constant of Variation" is known, finding these values becomes an easy task through algebraic manipulation.Going back to our previous scenario involving sales tax, suppose you're given values of \( y = 5.10 \) but not \( x \). With \( k = 0.06 \) (as calculated before), you can find this unknown value. Using the equation for direct variation, \( y = kx \), and plugging in the known values gives us \( 5.10 = 0.06x \).To solve for \( x \), rearrange the equation by dividing both sides by \( k \), leading to:\[x = \frac{5.10}{0.06}\]Calculating this gives \( x = 85 \). Therefore, a purchase that results in a \(5.10 sales tax corresponds to spending \)85. This process helps in finding any missing values once \( k \) is determined, ensuring accurate calculations.

Algebraic Equations

Algebraic equations are essential tools in mathematics, allowing us to represent relationships and solve for unknowns in a structured way. In scenarios involving direct proportionality, these equations describe how two variables relate.Consider the equation \( y = kx \) representing direct proportionality. It is an algebraic equation where \( y \) and \( x \) are variables, and \( k \) is a constant. This equation helps us express the correlation between spending and sales tax in our example. When given any two of these quantities, we can solve for the third.Working with such equations involves typical algebraic techniques:

• Substitution: Plug known values into the equation to find the unknown.
• Isolation: Rearrange the equation to isolate and solve for the desired variable.

Through these methods, algebraic equations provide a clear and straightforward pathway to finding solutions and understanding relationships between different quantities. In our sales tax scenario, algebraic equations made it possible to calculate the Constant of Variation, verify the relationship, and find any missing values efficiently.