Problem 139 - College Algebra with Modeling and Visualization

Question

Approximate the constant of variation to the nearest hundredth. The variable \(z\) varies jointly as the second power of \(x\) and the third power of \(y\). When \(x=2\) and \(y=2.5, z=31.9\).

Step-by-Step Solution

Answer

The constant of variation \( k \) is approximately 0.51.

Step 1: Understanding Joint Variation

In a joint variation scenario, a variable varies with the product of powers of other variables. This problem states that the variable \( z \) varies jointly as the square of \( x \) and the cube of \( y \). We can express this relationship as: \( z = kx^2y^3 \), where \( k \) is the constant of variation we need to find.

Step 2: Substitute Given Values

We are given values of \( x = 2 \), \( y = 2.5 \), and \( z = 31.9 \). Substitute these values into the equation: \( 31.9 = k(2)^2(2.5)^3 \).

Step 3: Simplify the Equation

Calculate the powers and products: \( 2^2 = 4 \) and \( (2.5)^3 = 15.625 \). Substitute these into the equation to get \( 31.9 = k imes 4 imes 15.625 \).

Step 4: Solve for the Constant of Variation

Simplify the equation: Compute the product \( 4 imes 15.625 = 62.5 \). Thus, \( 31.9 = 62.5k \). To isolate \( k \), divide both sides by 62.5: \( k = \frac{31.9}{62.5} \).

Step 5: Calculate and Round the Value of k

Perform the division \( \frac{31.9}{62.5} \approx 0.51 \). Round \( k \) to the nearest hundredth.

Key Concepts

Constant of Variation

The constant of variation, often denoted by the letter \( k \), is a key component in understanding relationships between variables in joint variation problems. It represents a fixed number that connects the variables through a specified algebraic equation. In the context of joint variation, the constant \( k \) determines the strength and nature of the relationship.To find the constant of variation, we isolate \( k \) in the equation by substituting known values of the variables and then solving for \( k \). In our given problem, we start with the relationship: \( z = kx^2y^3 \). We plug in the known values \( x = 2 \), \( y = 2.5 \), and \( z = 31.9 \) into the equation and simplify to isolate \( k \). Once calculated, \( k \) provides us with a more precise understanding of how changes in \( x \) or \( y \) will influence \( z \).

• The constant helps in predicting the behavior of variables.
• It remains unchanged for any specific problem setup once determined.
• Accurate calculation of \( k \) is crucial for precise results.

Algebraic Equations

Algebraic equations are mathematical statements that show the equality between two expressions. In problems involving variations, these equations express complex relationships systematically. Understanding how to manipulate and solve them is crucial in variation problems.For joint variation problems, the algebraic equation takes the form of \( z = kx^my^n \). This allows us to capture the relationship between multiple variables and their powers. Each term in the equation has a significant impact:

• \( z \) is the dependent variable whose value we want to determine or understand.
• \( x \) and \( y \) are independent variables, and their values influence \( z \).

By rearranging, substituting, and solving these equations, we can uncover the constant of variation, enabling us to make predictions. It's like unraveling a puzzle to see how each piece (or variable) fits into the bigger picture.

Powers of Variables

In joint variation, the powers of variables signify how strongly a variable affects the outcome. For example, in the formula \( z = kx^2y^3 \), the powers indicate how changes in \( x \) or \( y \) influence \( z \).

• The power of 2 for \( x \) (written as \( x^2 \)) means that if \( x \) doubles, \( z \) will quadruple, assuming \( y \) remains constant.
• The power of 3 for \( y \) (written as \( y^3 \)) means that if \( y \) doubles, \( z \) increases eightfold, assuming \( x \) remains constant.

These powers highlight the sensitivity of \( z \) to changes in \( x \) or \( y \). Larger powers result in larger impacts, requiring careful consideration when analyzing variations. Whether in physics, chemistry, or economics, understanding the role of powers can provide insights into complex systems and enhance problem-solving skills.