Problem 7 - College Algebra with Corequisite Support

Question

For the following exercises, write an equation describing the relationship of the given variables. \(y\) varies directly as the cube of \(x\) and when \(x=36, \quad y=24\).

Step-by-Step Solution

Answer

The equation is \(y = \frac{1}{1944}x^3\).

Step 1: Understanding Direct Variation

The statement "\(y\) varies directly as the cube of \(x\)" means that \(y\) is proportional to \(x^3\). The mathematical expression for this direct variation is: \( y = kx^3 \), where \(k\) is the constant of proportionality.

Step 2: Substitute Known Values

We are given that when \(x = 36, y = 24\). To find \(k\), substitute \(x = 36\) and \(y = 24\) into the equation. This gives us: \(24 = k(36^3)\).

Step 3: Calculate \(36^3\)

Calculate the cube of 36: \(36^3 = 36 \times 36 \times 36 = 46656\).

Step 4: Solve for the Constant \(k\)

Substitute \(36^3 = 46656\) back into the equation: \(24 = k \cdot 46656\). Simplify to solve for \(k\): \(k = \frac{24}{46656}\).

Step 5: Simplify the Constant \(k\)

Simplify the fraction: \( \frac{24}{46656} = \frac{1}{1944} \). Thus, \(k = \frac{1}{1944}\).

Step 6: Write the Final Equation

Now we have the constant \(k\), substitute \(k\) back into the original equation: \(y = \frac{1}{1944}x^3\). This is the equation describing the relationship between \(y\) and \(x\).

Key Concepts

Constant of Proportionality

The constant of proportionality, often represented by \( k \), is a fundamental concept in understanding proportional relationships in mathematics. It represents the factor by which one variable changes in relation to another.

In the context of direct variation, if \( y \) varies directly with the cube of \( x \), the relationship can be expressed as \( y = kx^3 \). Here, \( k \) determines the rate at which \( y \) changes as \( x \) changes.

• The equation \( y = kx^3 \) tells us that for every increase or decrease in \( x \), \( y \) changes proportionally based on the cube of \( x \) multiplied by \( k \).
• Without the constant, a simple, scalar relationship wouldn't exist between the variables.

Finding the constant of proportionality involves isolating \( k \) in the equation by substituting known values of \( x \) and \( y \) and solving for \( k \). This constant remains the same across the equation as long as the relationship is linear and proportional.

Cube Function

A cube function is a polynomial of the form \( x^3 \). It's important in mathematics because it naturally extends linear relationships to three dimensions.

In the context of direct variation, cubing introduces a non-linear relationship between variables. For example, if \( y \) depends directly on \( x^3 \), it means any change in \( x \) has an exponentially greater effect on \( y \) due to the cubic form.

• When you cube a number \( x \), you are multiplying \( x \) by itself three times: \( x \times x \times x \).
• This operation significantly increases the result, which highlights why the cube function causes \( y \) to grow much faster as \( x \) increases.

The cube function is essential for modeling real-world phenomena where growth or change occurs in three-dimensional space or follows a steep trajectory.

Proportional Relationships

Proportional relationships in mathematics explain how two quantities change in relation to each other. When two quantities are in a direct relationship, one is a constant multiple of the other.

Key characteristics of proportional relationships include:

• A constant ratio, or constant of proportionality, exists between the two variables.
• Their graph is a straight line that passes through the origin when dealing with linear forms.
• In direct variation, as one quantity increases, the other increases at a constant rate defined by the constant \( k \).

Understanding proportional relationships helps in identifying how variables are linked and predicting the outcomes based on these links. This is crucial in many fields such as physics, economics, and engineering where systems and components operate under proportional conditions.