Problem 91 - A Graphical Approach to College Algebra

Question

Assume that the constant of variation is positive. Suppose \(y\) varies directly with the third power of \(x .\) If \(x\) triples, what happens to \(y ?\)

Step-by-Step Solution

Answer

\( y \) becomes 27 times its original value.

Step 1: Understanding Direct Variation

In direct variation, if one variable changes, the other changes proportionally. The relationship is often described as \( y = k imes x^n \), where \( k \) is the constant of variation, and \( n \) the power to which \( x \) is raised.

Step 2: Identify the Given Relationship

We are told that \( y \) varies directly with the third power of \( x \), so the equation becomes \( y = k imes x^3 \). This means if \( x \) changes, \( y \) will change in direct relation to the cube of that change.

Step 3: Determine the Effect of Tripling \( x \)

Since \( x \) triples, this can be expressed as \( x' = 3x \). We substitute this into the relationship: \( y' = k imes (3x)^3 \).

Step 4: Calculate the Effect on \( y \)

Calculate \( (3x)^3 = 3^3 imes x^3 = 27x^3 \). Substituting back gives \( y' = k imes 27x^3 = 27y \).

Step 5: Conclusion

If \( x \) triples, \( y \) will become 27 times its original value.

Key Concepts

Power Functions

Power functions are mathematical expressions where a variable is raised to a specific exponent. In the general form, a power function can be expressed as \( y = k \times x^n \), where:\

\
• \( y \) is the dependent variable.
\
• \( k \) is the constant of variation, which remains the same across the function.
\
• \( x \) is the base variable.
\
• \( n \) is the exponent or power.
\

\In the context of direct variation problems, power functions help determine how changes in the variable \( x \) affect the variable \( y \). If the power \( n \) is positive, \( y \) increases as \( x \) increases. For instance, if \( x \) is a positive number doubled within the function \( y = k \times x^3 \), \( y \) will become eight times larger, because doubling \( x \) means multiplying by \( 2^3 = 8 \). Understanding how powers work is crucial for solving problems involving direct variation and multiple variables.

Constant of Variation

The constant of variation \( k \) is a crucial part of any direct variation equation. It determines the rate at which the dependent variable changes with respect to the independent variable. In the equation \( y = k \times x^n \), the constant \( k \) ties \( x \) and \( y \) together.
Imagine \( k \) as a factor that scales the function. Regardless of changes in \( x \), \( k \) remains unchanged, offering consistency across calculations. For example, in the equation \( y = 5x^3 \), the 5 is the constant of variation. This function tells us that for every unit change in \( x^3 \), \( y \) changes by 5 times that amount. This constant is what makes the relationship between \( x \) and \( y \) linear in the context of their direct variation.

Proportional Relationships

Proportional relationships emerge when two variables change at a constant rate relative to each other. In mathematical terms, a proportional relationship prompts an equation like \( y = k \times x^n \), indicating that any change in \( x \) results in a predictable change in \( y \) based on the consistent factor \( k \).
An easy way to understand this is to consider a simple example: travel distance. If a car travels at a constant speed, the distance traveled is directly proportional to the time spent driving. If speed, the constant of variation, is maintained, then doubling the driving time results in doubling the distance.When it comes to power functions, proportional relationships can be linear or non-linear depending on the power \( n \). If \( n = 1 \), the relationship is linearly proportional. But if \( n > 1 \), like in \( y = k \times x^3 \), tripling \( x \) results in \( y \) being 27 times the original amount, demonstrating a proportional relationship dictated by the cubic change of \( x \). This elucidates the essential nature of understanding power and constant of variation together in solving direct variation problems.