Problem 32 - College Algebra

Question

A person's wages, \(W,\) vary directly as the number of hours worked, \(h\) a. Write an equation that expresses this relationship. b. For a 40 -hour work week, Gloria earned \(\$ 1400\). Substitute 1400 for \(W\) and 40 for \(h\) in the equation from part (a) and find \(k,\) the constant of variation. c. Substitute the value of \(k\) into your equation in part (a) and write the equation that describes Gloria's wages in terms of the number of hours she works. d. Use the equation from part (c) to find Gloria's wages for 25 hours of work.

Step-by-Step Solution

Answer

The equation expressing the relationship is \(W = kh,\) the constant of variation is 35, and Gloria's wage equation is \(W = 35h.\) For 25 hours of work, Gloria will earn \$875.

Step 1: Writing the Equation

If the wages, \(W,\) vary directly as the number of hours worked, \(h,\) the equation for this direct variation is \(W = kh,\) where \(k\) is the constant of variation.

Step 2: Finding Constant of Variation

For a 40-hour work week, Gloria earned \$1400, which means we can plug these values into our equation and solve for \(k.\) So, \(1400 = k \cdot 40\) which simplifies to \(k = \frac{1400}{40}\), meaning \(k = 35\)

Step 3: Writing Specific Equation

Now substitute the value of \(k\) into your equation from Step 1. After substitution, the equation will be \(W = 35 \cdot h\). This equation will determine Gloria's wage based on the number of hours she works.

Step 4: Calculating Wages for Specific Hours

Using the equation from Step 3, we can calculate Gloria's wages for 25 hours of work. Substituting \(h = 25\) into our equation, we get \(W = 35 \cdot 25,\) which simplifies to \(W = \$875.\) Thus, Gloria will earn \$875 for 25 hours of work.

Key Concepts

Constant of Variation

The constant of variation, often denoted as \( k \), is a crucial factor in direct variation equations. In direct variation, one variable changes consistently with another, maintaining a proportional relationship. Here, the equation \( W = kh \) describes how wages \( W \) vary in relation to hours worked \( h \). The constant \( k \) determines how the wage scales with each hour of work.

To find \( k \), you need a specific example. For instance, if someone earns \(1400 for a 40-hour week, you can insert these values into the equation to solve for \( k \):

• Equation: \( 1400 = k \cdot 40 \)
• Solve for \( k \): \( k = \frac{1400}{40} \), which simplifies to \( k = 35 \)

This means that for every hour worked, the wages increase by \)35. Thus, the constant of variation \( k \) is essentially the wage per hour.

Linear Equations

Linear equations represent relationships with a constant rate of change and are foundational in understanding direct variation. In essence, a linear equation is any equation that graphs as a straight line. The direct variation equation \( W = kh \) is a linear equation. It reflects a straight-line relationship between wages and hours worked.

Linear equations can be expressed in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. However, in the context of direct variation, the equation simplifies because the graph always passes through the origin, and the intercept \( b \) is zero:

• Direct variation form: \( y = mx \)
• Here, \( m \) (the slope) is the constant of variation \( k \)

This means the equation \( W = 35h \) describes Gloria's wages as a linear function of hours worked.

Variable Relationships

Understanding variable relationships is essential in direct variations as they highlight how one variable affects another. In the context of wages and hours, this relationship is straightforward: as the number of hours worked increases, wages increase proportionally by the constant of variation \( k \).

Such relationships are common in real-world scenarios where two variables maintain a consistent ratio. Since \( W \) directly depends on \( h \), any change in \( h \) will predictably change \( W \). For example:

• If Gloria works double the hours, her wages will also double, provided \( k \) remains constant.
• The formula \( W = 35h \) helps predict her income based solely on the hours reported.

This direct proportionality means that if she works 25 hours, calculate \( W = 35 \cdot 25 \) to find her earning, which is $875. This shows how important it is to understand the underlying linear relationship and the impact of the constant of variation.