Problem 101 - College Algebra with Modeling and Visualization

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understanding Direct Variation with Powers

When a variable \( y \) varies directly as the third power of another variable \( x \), it means \( y = kx^3 \) where \( k \) is the positive constant of proportionality.

Step 2: Considering the Effect of Tripling \( x \)

If \( x \) triples, it means we replace \( x \) with \( 3x \). The equation becomes \( y = k(3x)^3 \).

Step 3: Calculating the New Expression

Substitute \( 3x \) into the equation to get \( y = k(3x)^3 = k imes 27x^3 = 27kx^3 \).

Step 4: Comparing with the Original Expression

Originally, \( y = kx^3 \). With \( x \) tripled, the expression for \( y \) becomes \( y' = 27kx^3 \). This shows that \( y' = 27y \), meaning \( y \) is 27 times its original value.

Key Concepts

Proportionality Constant

In mathematics, the proportionality constant is a factor that scales a quantity in direct variation relationships. Essentially, it is the "k" in equations like

• \( y = kx^3 \)

which represents the constant of proportionality when \( y \) is directly proportional to the cube of \( x \).
This constant is crucial because it dictates the specific relationship between \( x \) and \( y \) for a given problem.
When you encounter equations involving direct variation, the proportionality constant ensures that the relationship between the two variables is correctly balanced and maintains their proportional connection.
It remains unchanged as long as the nature of the relationship does not change. By understanding the role of the proportionality constant, you can predict or calculate changes in one variable when the other variable changes.

Power of a Variable

In the exercise provided, the variable \( y \) varies directly as the third power of \( x \). This implies that the relationship between \( y \) and \( x \) involves exponentiation, specifically to the power of three:

• \( y = kx^3 \)

The "power of a variable" in mathematical terms refers to the exponent applied to the variable. In our equation, the third power signifies that \( x \) is cubed.
This impacts how \( y \) changes in relation to \( x \). Whenever raising or updating values, the power indicates how many times you multiply the variable by itself.
The greater the power, the more dramatically changes in \( x \) will affect \( y \). Understanding powers is fundamental when dealing with direct variation, and in this case, it highlights the dynamic impact of changes in \( x \) on \( y \).

Triple Effect on Variables

When \( x \) triples in the equation \( y = kx^3 \), it significantly escalates the effect on \( y \). This is because both the direct variation and the power at which \( x \) is raised play pivotal roles.
Let's analyze what happens step by step:
If \( x \) becomes \( 3x \), substituting this into our equation will yield:

• \( y = k(3x)^3 \)

Calculating further gives us:

• \( y = k \times 27x^3 \)
• Thus: \( y = 27kx^3 \)

Here, we see that \( y \) increases by 27 times its original value, that is, \( y' = 27y \).
This illustrates how, in direct variation with powers, increases in the independent variable (\( x \)) can cause exponential increases in the dependent variable (\( y \)).
The tripling of \( x \) combined with the cubic power dramatically amplifies its impact on \( y \). Understanding this principle is essential for predicting outcomes in similar mathematical scenarios.