Problem 31 - College Algebra

Question

For the following exercises, use the given information to find the unknown value. \(y\) varies inversely with the cube of \(x\). When \(x=3\) then \(y=1\). Find \(y\) when \(x=1\).

Step-by-Step Solution

Answer

When \( x = 1 \), \( y = 27 \).

Step 1: Understand the Relationship

Since 'y varies inversely with the cube of x', this means the relationship can be expressed as \( y = \frac{k}{x^3} \), where \( k \) is the constant of variation.

Step 2: Find the Constant of Variation

We are given that when \( x = 3 \), \( y = 1 \). Substitute these values into the equation to find \( k \): \( 1 = \frac{k}{3^3} \). This simplifies to \( 1 = \frac{k}{27} \). Solving for \( k \), we get \( k = 27 \).

Step 3: Substitute and Solve for y

Now that we know \( k = 27 \), we substitute it back into the inverse variation equation: \( y = \frac{27}{x^3} \). We need to find \( y \) when \( x = 1 \). Substitute \( x = 1 \) into the equation: \( y = \frac{27}{1^3} = 27 \).

Step 4: Verify the Solution

Verifying our steps, the calculated \( y \) when \( x = 1 \) should be consistent with the equation and initial conditions. Since \( y = \frac{27}{x^3} \) held true for \( (x=3, y=1) \), and consistent substitutions provided \( y = 27 \) for \( x = 1 \), our solution is verified correct.

Key Concepts

Constant of Variation

The constant of variation plays a crucial role in understanding relationships between variables in inverse variation equations. The constant, usually denoted as \( k \), remains unchanged for a particular relationship. It helps in formulating how one variable affects another inversely. If you see a sentence like "\( y \) varies inversely with the cube of \( x \)," it implies: \( y = \frac{k}{x^3} \). Here:

• \( y \) is inversely related to the cube of \( x \).
• \( k \) remains constant across different values of \( x \) and \( y \).

Understanding the constant of variation is key to solving and verifying inverse variation problems. In this exercise, once \( k \) was found to be 27 for the given condition \((x=3, y=1)\), it applied uniformly when \( x \) changed to 1.

Cube of a Number

Cubing a number involves raising it to the power of three. It is calculated as \( x \times x \times x \), or \( x^3 \). Cubing:

• Grows the value exponentially, making small numbers very small and larger ones much bigger.
• Is essential in inverse relationships, especially when one variable inversely depends on another's cube.

For instance, in the problem, we calculated \( 3^3 \) to find that \( 27 \) influenced the value of \( y \). Remembering this concept helps in visualizing how alterations in \( x \) affect associated outcomes in inverse equations.

Algebraic Equations

Algebraic equations are mathematical statements that use symbols and variables to express relationships. In `inverse variation`, they allow us to set up formulas representing how variables interact:

• The structure \( y = \frac{k}{x^3} \) is derived from understanding the relationship as \( y \) inversely varies with \( x^3 \).
• Solving these equations typically involves isolating one variable by substituting known values and determining others.

In the given exercise, we first found \( k \) by substituting the initial conditions \((x=3, y=1)\), then solved for \( y \) using the determined \( k \) value. Mastery of such algebraic methods ensures accurate problem-solving across various mathematical challenges.