Problem 43 - Intermediate Algebra

Question

Express each verbal model in symbols. \(P\) varies directly as the square of \(a\) and inversely as the cube of \(j\)

Step-by-Step Solution

Answer

The symbolic representation is \( P = \frac{k \times a^2}{j^3} \).

Step 1: Identify the Relationship

To express the relationship verbally given in mathematical terms, start by identifying the type of variation. Here, it states that variable \(P\) varies directly with the square of another variable \(a\) and inversely with the cube of \(j\). This means we need to translate these directly into a mathematical equation.

Step 2: Translate Direct Variation

In a direct variation, the variable \(P\) is a constant multiple of the square of \(a\). If we let \(k\) be the constant of proportionality, this part of the equation becomes \(P = k \times a^2\).

Step 3: Translate Inverse Variation

The inverse variation states that \(P\) is also inversely proportional to \(j^3\), meaning \(P\) is divided by the cube of \(j\). Thus, the full equation becomes \(P = \frac{k \times a^2}{j^3}\).

Step 4: Combine Both Variations

Combine both variations from the previous steps into a single equation to properly express the entire relationship: \( P = \frac{k \times a^2}{j^3} \). This encompasses both the direct and inverse variations described in the problem.

Key Concepts

Proportionality Constant

In problems involving direct and inverse variation, the proportionality constant, typically represented by \(k\), plays a crucial role. It acts as a bridge connecting two or more variables, ensuring that their relationship is consistent.
Consider a situation where a variable \(P\) varies directly as the square of another variable \(a\). This relationship is represented as \(P = k \, a^2\). Here, \(k\) signifies how much \(P\) changes when \(a\) changes.
Similarly, the inverse relationship involves dividing by a power of another variable. For \(P\) that varies inversely with the cube of \(j\), we have \(P = \frac{k}{j^3}\).
Understanding the proportionality constant helps in predicting how one variable affects another in complex equations.

Mathematical Equations

Equations are tools that allow us to represent relationships between variables in a precise way. They are like sentences in the language of mathematics.
In our exercise, the relationship is presented through two types of equations. The direct variation equation is \(P = k \, a^2\). This suggests that \(P\) and \(a\) are directly proportional through a constant \(k\).
The inverse variation equation adds complexity: \(P = \frac{k \, a^2}{j^3}\). This equation shows that \(P\) is inversely related to \(j\). As \(j\) increases, \(P\) decreases, assuming \(a\) stays the same.
Together, these equations describe how multiple variables interact and influence each other simultaneously.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operators. They form the backbone of algebraic manipulation and problem-solving.
In the given context, the expression \(k \, a^2\) represents the direct variation part, showing \(a\)'s influence on \(P\). It encapsulates multiplication of the constant \(k\) and the square of \(a\).
The combined algebraic expression \(\frac{k \, a^2}{j^3}\) integrates the effects of both \(a\) and \(j\) on \(P\). This expression illustrates the inverse relationship with \(j^3\), highlighting how algebra communicates complex relationships succinctly.
To master algebra, it’s essential to be fluent in manipulating and interpreting these expressions, recognizing how each variable and constant affects the overall expression.