Problem 24 - Elementary and Intermediate Algebra

Question

Express each verbal model in symbols. See Objectives 3 and 4. \(M\) varies inversely as the cube of \(n\) and jointly as \(x\) and the square of \(z\)

Step-by-Step Solution

Answer

The equation is \( M = k \cdot \frac{xz^2}{n^3} \).

Step 1: Understand the Verbal Model

The exercise asks us to translate a verbal model into a mathematical expression. The statement 'M varies inversely as the cube of n and jointly as x and the square of z' implies an equation structure where a variable is both part of a joint variation and an inverse variation.

Step 2: Express Inverse Variation

Inverse variation indicates that one variable is inversely proportional to another variable raised to some power. Here, M varies inversely as the cube of n, so we can express this as \( M \propto \frac{1}{{n^3}} \).

Step 3: Express Joint Variation

Joint variation means that a variable is directly proportional to the product of two or more other variables. According to the problem, M also varies jointly as x and the square of z, which we can write as \( M \propto xz^2 \).

Step 4: Combine the Variations

Combine the inverse variation from Step 2 and the joint variation from Step 3 into a single equation. Since M is both inversely proportional to the cube of n and directly proportional to the product of x and the square of z, we can write: \( M = k \cdot \frac{xz^2}{n^3} \), where k is the constant of proportionality.

Step 5: Write the Complete Equation

Finalize the expression by taking both variations into account: The equation representing the model is \( M = k \cdot \frac{xz^2}{n^3} \). This completes the translation from the verbal model to a symbolic equation.

Key Concepts

Joint Variation

Joint variation is when a variable depends on two or more other variables in a multiplicative way. Each variable contributes to the change in the dependent variable. In our problem, the variable \( M \) varies jointly as \( x \) and \( z^2 \). This means that as \( x \) or \( z \) increases, \( M \) also increases, assuming the other variables stay constant.

• Joint variation can be expressed using direct proportionality between the dependent variable and the product of multiple variables.
• For instance, if \( M = k \, xz^2 \), this form shows \( M \) is directly related to both \( x \) and \( z^2 \).

To understand it clearly, if both \( x \) and \( z \) were to double, the effect on \( M \) would be straightforward as it multiplies with both variables.

Proportionality

Proportionality is a key concept in understanding variations. It refers to a constant relationship between two variables. If you say one quantity is proportional to another, changes in one quantity cause changes in the other by a constant factor. There are different types of proportionality:

• **Direct proportionality**: When one variable increases, the other increases by the same factor. \( y \propto x \) implies \( y = kx \).
• **Inverse proportionality**: When one variable increases, the other decreases, following a reciprocal relation. For example, in our exercise, \( M \propto \frac{1}{n^3} \) shows \( M \) decreases as \( n \) increases.

These principles of proportionality help us convert verbal problems into mathematical expressions, allowing the clear use of equations for problem-solving.

Mathematical Expressions

Mathematical expressions are concise representations of mathematical ideas using symbols and numbers. They play a crucial role in translating verbal statements into clear, usable forms. When dealing with complex ideas like inverse and joint variation, mathematical expressions can summarize the relationships.

• **Using expressions**: Enables quick calculations and simplifications in problem-solving.
• **Symbols**: Such as \( \propto \) for proportionality, or \( = \) for equality, define exact relationships.
• **Constants**: Like \( k \), embody fixed values that remain unchanged across scenarios.

Converting a verbal model into a mathematical expression, such as \( M = k \cdot \frac{xz^2}{n^3} \), provides a structured way to understand the dynamics between different variables.