Problem 33 - College Algebra

Question

For the following exercises, use the given information to find the unknown value. \(y\) varies inversely with the cube root of \(x\). When \(x=27,\) then \(y=5 .\) Find \(y\) when \(x=125 .\)

Step-by-Step Solution

Answer

When \(x=125\), \(y=3\).

Step 1: Understanding Inverse Variation

Inverse variation means that as one value increases, the other decreases such that their product remains constant. In this case, \(y\) varies inversely with the cube root of \(x\), represented as \(y \propto \frac{1}{\sqrt[3]{x}}\).

Step 2: Establishing the Equation

To form an equation, introduce a proportionality constant \(k\). Hence, \(y \cdot \sqrt[3]{x} = k\). With the information given when \(x=27\), \(y=5\), substitute these values into the equation: \(5 \cdot \sqrt[3]{27} = k\).

Step 3: Solve for the Constant

Calculate \(\sqrt[3]{27}\), which is \(3\). Therefore, the equation becomes \(5 \times 3 = k\), which simplifies to \(k = 15\).

Step 4: Using the Constant for New Conditions

To find \(y\) when \(x=125\), use the equation \(y \cdot \sqrt[3]{125} = 15\). Calculate \(\sqrt[3]{125}\), which is \(5\).

Step 5: Solve for New y Value

Substitute \(\sqrt[3]{125} = 5\) back into the equation to get \(y \times 5 = 15\). Solve for \(y\) by dividing both sides by \(5\), giving \(y = 3\).

Key Concepts

Cube Roots

Cube roots are an essential concept in algebra, especially when dealing with equations involving inverse variation. The cube root of a number is a value that, when multiplied by itself three times, gives the original number. For example, the cube root of 27 is 3, because \(3 \times 3 \times 3 = 27\). When solving problems where the variable is involved in cubic form, you will often need to

• Calculate cube roots of given numbers.
• Recognize patterns, such as \(\sqrt[3]{x}\).
Understanding cube roots helps simplify complex algebraic expressions, making problem-solving more straightforward.

Proportionality Constant

In inverse variation problems, you'll often use a proportionality constant, denoted as \(k\). This constant helps you form an equation from the relationship described by the problem. The equation

• Shows the connection between two variables, like \(y \propto \frac{1}{x}\).
• Identifies how changes in one variable affect the other.

In our specific exercise, the constant \(k\) helps to define the inverse relationship: \(y \cdot \sqrt[3]{x} = k\). By knowing \(k\), you can solve for an unknown value when the other variable changes.

Algebraic Equations

Algebraic equations serve as the backbone of mathematical problem-solving. They allow you to express relationships and unknowns in simple, understandable terms. Using the example given:

• The inverse relationship was captured by the equation \(y \cdot \sqrt[3]{x} = k\).
• Substitution of known values allows you to solve for \(k\).

Algebraic manipulation, such as isolating \(y\) or \(x\), is crucial to finding solutions. Ensure every step keeps the equation balanced, paying attention to units and variable placement.

Problem Solving Steps

Solving any mathematical equation requires a set of clear, concise steps. Breaking down a problem into manageable parts makes it easier to solve. Consider the following process:

• Identify the type of relationship, such as inverse variation.
• Set up the equation using a proportionality constant.
• Solve for the constant using known values.
• Apply the constant to new conditions to find the unknown value.

This step-by-step approach not only builds your understanding but improves your confidence in tackling similar problems.