Problem 31 - Elementary and Intermediate Algebra

Question

Express each variation model in words. In each equation, is the constant of variation. See Objectives 3 and 4. $$ P=\frac{k m}{n} $$

Step-by-Step Solution

Answer

\( P \) is directly proportional to \( m \) and inversely proportional to \( n \).

Step 1: Identify the Type of Variation

The equation given is \( P = \frac{k m}{n} \). This formula suggests a joint variation because \( P \) varies directly with \( m \) and inversely with \( n \). Joint variation occurs when a variable depends on two or more other variables, combining direct and inverse variations.

Step 2: Express Direct Variation in Words

Identify the part of the equation that shows direct variation. Here, \( P \) is directly proportional to \( m \), meaning when \( n \) is held constant, an increase in \( m \) will result in an increase in \( P \) by the same factor. In words: \( P \) varies directly with \( m \).

Step 3: Express Inverse Variation in Words

Identify the part of the equation that shows inverse variation. Here, \( P \) varies inversely with \( n \), meaning when \( m \) is constant, an increase in \( n \) will lead to a decrease in \( P \) while a decrease in \( n \) will increase \( P \). In words: \( P \) varies inversely with \( n \).

Step 4: Combine the Variations in a Sentence

Combine the descriptions from Steps 2 and 3 into a full statement that describes the variation model. The correct expression in words for the equation \( P = \frac{k m}{n} \) is: \( P \) is directly proportional to \( m \) and inversely proportional to \( n \).

Key Concepts

Direct Variation

Direct variation is a concept where one variable changes directly with another. That means when one variable increases, the other also increases at a constant rate.
In mathematical terms, if a variable \( y \) changes directly with \( x \), we say \( y \) is directly proportional to \( x \), and this is written as \( y = kx \) where \( k \) is the constant of proportionality.
In practical terms, direct variation is similar to everyday experiences. Imagine blowing air into a balloon. More air increases the size of the balloon. This is like saying the size is directly proportional to the amount of air.

• If \( x \) doubles, \( y \) doubles.
• If \( x \) is halved, \( y \) is halved too.

In the exercise, \( P \) directly varies with \( m \). This means, as \( m \) increases, \( P \) increases at the same rate when \( n \) is constant.

Inverse Variation

In inverse variation, one variable increases while the other decreases. They are inversely related, which means their product is constant. If a variable \( y \) varies inversely with \( x \), we write \( y = \frac{k}{x} \), where \( k \) is a constant.
Think of a seesaw. As one side goes up, the other comes down. This motion is akin to inverse variation.

• If \( x \) doubles, \( y \) becomes half.
• If \( x \) is halved, \( y \) doubles.

In our exercise, \( P \) inversely varies with \( n \). This tells us that if \( n \) increases, \( P \) decreases, given that \( m \) remains fixed. It is like sharing a pizza among friends; more friends mean each person gets less.

Variation Models

Variation models combine different types of variation relationships in mathematical expressions. Often, real-world scenarios require understanding how multiple variables interact, beyond just direct or inverse relationships.
A joint variation model, such as \( P = \frac{k m}{n} \), blends both direct and inverse variations. Here, \( P \) depends directly on \( m \) and inversely on \( n \).
By understanding such models, we interpret how changes in one part affect the whole. It’s like managing resources where increasing one resource might need reducing another to maintain balance.
The power of variation models lies in their utility:

• They describe complex relationships simply.
• They help predict outcomes when changing variables.

Mastering these concepts enables solving a range of mathematical and real-world problems efficiently.