Problem 44 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identifying Keywords

First, identify the keywords from the verbal description. The phrase "M varies inversely as the cube of n" indicates an inverse relationship, while "jointly as x and the square of z" refers to a joint direct variation.

Step 2: Inverse Variation Setup

Express the inverse variation part of the model. Since M varies inversely as the cube of n, this is represented by: \[ M \propto \frac{1}{{n^3}} \]

Step 3: Direct Joint Variation Setup

For the joint variation, since M varies directly as x and the square of z, this portion of the variation is expressed by: \[ M \propto xz^2 \]

Step 4: Combining Variations

Combine the inverse and joint variation expressions. Since M varies both inversely with respect to n and jointly with x and the square of z, combine them into one equation: \[ M = k \cdot \frac{xz^2}{n^3} \] Here, \( k \) is a constant of proportionality.

Key Concepts

Joint Variation

Joint variation occurs when one quantity depends on the product of two or more other quantities. It's the scenario where a variable varies directly in relation to more than a single variable. Consider joint variation as a multi-way partnership between variables. For instance, in our exercise, the variable \( M \) is said to vary directly with both \( x \) and \( z^2 \). This forms the basis for a joint variation expression such as \( M \propto xz^2 \).

Joint variation can be very useful when determining outcomes influenced by multiple factors all at once. This expression shows that \( M \) increases when either \( x \) increases, \( z^2 \) increases, or both. It captures the simultaneous effects of these influencing factors, encapsulating them into a single relational expression.

Direct Variation

Direct variation is a simpler form of the relationship where one variable increases at the same rate as another. The heart of direct variation is the idea that as one variable increases, another variable does so in a proportional manner. This is expressed by equations in the form of \( y = kx \), where \( k \) is the proportionality constant.

In our context, part of \( M \)'s variation is direct with respect to the variable \( x \) and the square of another variable \( z \). Hence, the equation representing only this part would be \( M \propto xz^2 \). Direct variation can help in predicting how changes in one variable can directly influence another, assuming that all else remains constant.

Proportionality Constant

The proportionality constant, commonly represented by \( k \), is a crucial element in both direct and inverse variation equations. It allows the expression of a precise relationship between variables rather than just a proportional tendency. In our exercise solution, \( k \) is part of the equation \( M = k \cdot \frac{xz^2}{n^3} \). Here, \( k \) quantifies the relationship's exactness.

The proportionality constant ties the elements of the equation together, enabling the prediction of actual numerical outcomes. Without it, equations would only show proportionality but not actual values. Finding \( k \) might involve initial conditions or given numerical values to define the specific circumstances of the scenario.