Problem 15 - College Algebra with Corequisite Support

Question

For the following exercises, write an equation describing the relationship of the given variables. \(y\) varies inversely as the cube root of \(x\) and when \(x=64, \quad y=5\).

Step-by-Step Solution

Answer

The equation is \( y = \frac{20}{\sqrt[3]{x}} \).

Step 1: Understanding Inverse Variation

Since \( y \) varies inversely as the cube root of \( x \), it means \( y \) is equal to a constant \( k \) divided by the cube root of \( x \). The initial equation we write is \( y = \frac{k}{\sqrt[3]{x}} \).

Step 2: Substitute Given Values

We know that when \( x = 64 \), \( y = 5 \). Substitute these values into the equation to find the constant \( k \): \( 5 = \frac{k}{\sqrt[3]{64}} \).

Step 3: Calculate the Cube Root

Compute the cube root of 64: \( \sqrt[3]{64} = 4 \). Now substitute back into the equation: \( 5 = \frac{k}{4} \).

Step 4: Solve for the Constant k

To find \( k \), multiply both sides by 4: \( 5 \times 4 = k \), hence \( k = 20 \).

Step 5: Write the Equation

Substitute \( k \) back into the inverse variation equation to express \( y \) as a function of \( x \): \( y = \frac{20}{\sqrt[3]{x}} \).

Key Concepts

Cube Root

A cube root is a number that, when multiplied by itself three times, gives the original number. It's the opposite operation of cubing a number. For example, the cube root of 64 is 4 because when you multiply 4 by itself three times (i.e., 4 × 4 × 4), you get 64. This concept is crucial when dealing with inverse variation problems involving cube roots.
The cube root function is represented as \( \sqrt[3]{x} \), where \( x \) is the number you're finding the cube root for. In algebra, it's important to understand how to compute this because it helps to solve equations where a variable is inversely proportional to a cube root.

• The function \( \sqrt[3]{x} \) computes the cube root of \( x \).
• A cube root scenario is often seen in physical problems involving volumes and capacities.
• Understanding cube roots helps in simplifying algebraic expressions with radical terms.

Algebraic Equations

Algebraic equations are mathematical statements that contain variables and constants. In the provided exercise, we deal with an equation in the form of inverse variation. Specifically, the equation of the form \( y = \frac{k}{\sqrt[3]{x}} \) describes a relationship where \( y \) changes inversely with the cube root of \( x \).
In algebra, understanding the setup of equations like this allows us to work out unknowns by performing operations such as substitution, multiplication, and division.

• Recognize different types of algebraic equations, such as linear, quadratic, and inverse variations.
• Use the given values to make substitutions in the equations to help solve unknowns.
• The ability to rearrange and solve equations is a crucial algebra skill.

Constant of Variation

In variation problems, the constant of variation, often denoted as \( k \), is a crucial element. It describes how two variables relate to one another. In an inverse variation problem, this constant is crucial for establishing how drastically one variable changes as the other variable changes.
For the inverse variation equation \( y = \frac{k}{\sqrt[3]{x}} \), the constant of variation \( k \) is calculated by substituting the known values into the equation and solving for \( k \). This step solidifies the relationship between the variables.

• Understanding the constant of variation is essential in describing the exact relationship in variation problems.
• Once \( k \) is determined, it can be used to predict other values of \( y \) for different values of \( x \).
• In the specific exercise, \( k \) was calculated as 20 by solving \( 5 = \frac{k}{4} \).

Problem Solving

Problem-solving in algebra involves applying mathematical concepts to find solutions to exercises or real-world problems. This specific problem showcases the steps needed to solve inverse variation relationships effectively.
The solution involves understanding the relationship between variables, using cube roots, and solving algebraic equations with a constant of variation.
To solve similar problems yourself:

• Start by understanding the problem statement thoroughly. Know what you need to find (solve for \( y \)).
• Translate the text into a mathematical equation. Recognize the type of variation present, in this case, inverse variation.
• Identify and substitute given values to find unknowns like \( k \).
• Re-check calculations at each step to ensure accuracy.
• Substitute back to find the general solution or specific predictions.

Effective problem-solving involves not just following steps, but also understanding each action and its purpose toward arriving at a solution.