Problem 14 - Intermediate Algebra

Question

Find the constant of variation for each of the stated conditions. \(y\) varies directly as the cube of \(x\), and \(y=48\) when \(x=\) \(-2 .\)

Step-by-Step Solution

Answer

The constant of variation is \(k = -6\).

Step 1: Understand Direct Variation

In this problem, since \(y\) varies directly as the cube of \(x\), we can express this relationship as \(y = kx^3\), where \(k\) is the constant of variation we need to find.

Step 2: Substitute the Given Values

Given \(y = 48\) and \(x = -2\), substitute these values into the equation from Step 1: \(48 = k(-2)^3\).

Step 3: Solve for the Constant of Variation

First, calculate \((-2)^3\), which is \(-8\). Now the equation becomes \(48 = k(-8)\). Solve for \(k\) by dividing both sides by \(-8\): \(k = \frac{48}{-8}\).

Step 4: Simplify to Find \(k\)

Simplifying \(\frac{48}{-8}\) gives \(k = -6\). Thus, the constant of variation \(k\) is \(-6\).

Key Concepts

Constant of Variation

When dealing with problems of direct variation, the concept of the constant of variation is vital. The constant of variation, often represented by the letter \(k\), establishes the fixed proportionality between two variables that vary directly. In the context of the problem, we have a relationship where \(y\) varies directly as the cube of \(x\). This means we can express it as:

• \(y = kx^3\)

Here, \(k\) is not just a random number; it provides a consistent measure of how \(y\) changes as \(x\) is altered.Different conditions or inputs will yield different constants of variation, but within each scenario, \(k\) remains ever faithful in its constancy.This consistency allows us to make predictions and understand relationships in a predictable manner. Providing the numerical value of \(k\), based on given data, solidifies our understanding of the direct relationship between variables.

Cube of a Number

The operation of cubing a number is straightforward, yet pivotal in many mathematical contexts, including direct variation.Cubing a number simply means raising it to the power of three. This is written as \(x^3\), implying the number multiplied by itself twice:

• For example, if \(x = -2\), then \((-2)^3 = (-2) \times (-2) \times (-2) = -8\).

Cubing is unique because it preserves the sign of negative numbers and increases the magnitude significantly faster than simple squaring or basic multiplication.Understanding the cube of a number allows one to grasp why certain direct variations, like in our example, involve much larger changes in output, relative to changes in input.This operation is a foundational skill, underpinning more complex algebraic manipulations like the ones encountered in solving for a constant of variation.

Substitution Method

The substitution method is a practical, often used technique in algebra that simplifies solving equations with known values.It involves replacing variables with their given numerical values, essentially customizing a general equation to fit specific data.In our problem, where \(y = 48\) and \(x = -2\), substituting these values into the direct variation equation \(y = kx^3\) becomes essential:

• Substitute \(y = 48\) and \(x = -2\) into \(y = kx^3\).
• Resulting in: \(48 = k(-2)^3\).

Using substitution helps break down the problem, moving it from abstract to concrete.This method is particularly important when trying to uncover values for unknowns, like our constant \(k\), providing a clear-cut pathway from problem statement to solution.It’s a method you’ll find useful in a variety of algebraic situations, ensuring clarity and precision in math problem-solving.