Problem 7 - College Algebra

Question

Use the four-step procedure for solving variation problems given on page 424 to solve. \(y\) varies jointly as \(x\) and \(z . y=25\) when \(x=2\) and \(z=5\) Find \(y\) when \(x=8\) and \(z=12\)

Step-by-Step Solution

Answer

The value of \( y \) when \( x = 8 \) and \( z = 12 \) is 240.

Step 1: Define Variation Formula

Joint variation means that \( y \) varies directly as the product of \( x \) and \( z \). Therefore, we can write the formula as \( y = kxz \) where \( k \) is the constant of variation.

Step 2: Compute the Constant of Variation

Use the provided values of \( x, z, \) and \( y \) to find \( k \). Plugging in the values: \( 25 = k * 2 * 5 \). By solving this equation, \( k = 2.5 \).

Step 3: Utilize New Values to Calculate Y

Now that we know the constant of variation \( k \), we can find \( y \) when \( x = 8 \) and \( z = 12 \). Substituting these values into the formula gives: \( y = 2.5 * 8 * 12 \).

Step 4: Evaluate Final Result

By evaluating the multiplication, we find that \( y = 240 \).

Key Concepts

Variation Formula

In mathematics, the concept of a variation formula is vital when dealing with relationships between different quantities. Joint variation describes a situation where one variable varies directly as the product of two or more other variables. In simpler terms, if one quantity increases or decreases, the other will do the same, maintaining a consistent product. The variation formula in the given problem is tied to joint variation, which combines separate elements of direct variation.

To express this relationship, we can use the equation: \[ y = kxz \] Here, \( y \) represents the variable that changes based on the values of \( x \) and \( z \). The letter \( k \) stands for the constant of variation, which maintains proportionality among \( y, x, \) and \( z \).

In this specific case, the goal is to understand how \( y \) changes in response to changes in \( x \) and \( z \). The formula helps in predicting or calculating one of these variables when the others are known, making it a powerful tool for solving real-world problems where such relationships occur.

Constant of Variation

Understanding the constant of variation, denoted as \( k \), is crucial in solving problems involving joint variation. This constant ensures that the relationship between the variables remains uniform across all instances. In our problem, by substituting the known values of \( y = 25, x = 2, \) and \( z = 5 \), we find \( k \) through the variation formula:

• Start with the formula: \( y = kxz \).
• Substitute the existing values: \( 25 = k \times 2 \times 5 \).
• Solving for \( k \), we find \( k = 2.5 \).

By discovering \( k = 2.5 \), we confirm how critical this constant is for predicting different scenarios using the same variation formula. With \( k \) fixed, even when \( x \) and \( z \) change, the calculated value of \( y \) will accurately reflect the relationship set by this constant.

Direct Variation

Direct variation is one of the key principles underpinning joint variation. In direct variation between two variables, if one variable changes, the other changes in the same proportion. Essentially, if one doubles, triples, or otherwise modifies its value, the other follows suit in direct proportion.

The joint variation problem combines this concept with two variables affecting a third. Specifically, the equation \( y = kxz \) relies on direct variation principles for both \( x \) and \( z \).

• Both \( x \) and \( z \) directly influence \( y \).
• Increase in either \( x \) or \( z \), assuming \( k \) and the other variable remain constant, will cause \( y \) to increase proportionally.
• The formula, thus, behaves like two direct variations acting together on \( y \).

Such a mapping gives a consistent and predictable model, useful in practical scenarios like physics or economics, where understanding the interplay between multiple factors is necessary.