Problem 91 - A Graphical Approach to Precalculus with Limits

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

If \(x\) triples, \(y\) becomes 27 times larger.

Step 1: Understand Direct Variation

When we say that "\(y\) varies directly with the third power of \(x\)," it means that \(y\) is proportional to \(x^3\). This can be written as \(y = kx^3\), where \(k\) is the constant of variation.

Step 2: Substitute Values for Tripling x

If \(x\) triples, then the new \(x\) becomes \(3x\). Substitute this into the equation \(y = kx^3\) to find the new expression for \(y\). Thus, \(y = k(3x)^3\).

Step 3: Calculate the New y Value

Now calculate \((3x)^3\). This is \(3^3 \cdot x^3 = 27x^3\). So, the expression becomes \(y = k \cdot 27x^3\). Therefore, the new \(y\) is \(27kx^3 = 27y\).

Step 4: Conclusions

If \(x\) triples, the new expression for \(y\) is \(27y\). Thus, \(y\) increases by a factor of 27 when \(x\) triples.

Key Concepts

Proportionality

In mathematics, proportionality refers to the relationship between two quantities where one quantity is a constant multiple of the other.
This means if one quantity changes, the other changes in a predictable way based on a constant ratio. This relationship is often described using a proportionality statement or equation.

For example, in direct variation, if two variables, say \( y \) and \( x \), are proportional, then there exists a constant \( k \) such that \( y = kx \).
Here, \( k \) is known as the constant of variation, linking the two quantities. In the context of the exercise, the direct variation is expressed as \( y = kx^3 \).

Remember, with direct variation, a change in one variable will directly affect the other.

• When \( x \) increases, \( y \) increases as long as \( k \) is positive.
• If \( x \) decreases, \( y \) also decreases.

This forms the backbone of many direct variation problems, providing a clear pathway to understanding changes in related variables.

Constant of Variation

The constant of variation, represented by \( k \) in equations, is a critical component in understanding direct variation.
It serves as the unchanging multiplier that relates two variables in a proportional relationship.

For the equation \( y = kx^3 \), \( k \) is the constant that maintains the proportionality between \( y \) and the cubic power of \( x \).
In this relationship, even if \( x \) changes, \( k \) remains the same, ensuring the proportionality between the two variables.

When solving for changes in \( y \) as affected by changes in \( x \), knowing \( k \) helps:

• If \( x \) is scaled by a certain factor, \( y \) is scaled by that factor raised to the power related to \( x \) in the relationship.
• Thus, if \( x \) is tripled, then \( (3)^3 \), which is 27, becomes the multiplication factor for \( y \).

The role of \( k \) is crucial in providing stability and predictability within these mathematical relationships.

Power Functions

Power functions represent relationships where a variable is raised to a certain power, influencing how changes to that variable impact the function’s output.
In the context of the given exercise, the power function is \( y = kx^3 \). This involves the variable \( x \) being raised to the third power, emphasizing the pronounced effect even small changes in \( x \) can have on \( y \).

Understanding power functions involves several points:

• The exponent determines the degree of change; in this case, as \( x \) is cubed, any increase or decrease in \( x \) dramatically influences \( y \).
• When \( x \) triples, the cubic power of \( x \) means \( y \) becomes \( 3^3 \times \) original \( y \), which equals 27 times the original \( y \).

Power functions appear frequently in scientific, economic, and engineering problems, where they model exponential growth or escalation, critical for analyzing trends and forecasting outcomes.