Problem 1 - Algebra and Trigonometry with Analytic Geometry

Question

Express the statement as a formula that involves the given variables and a constant of proportionality \(k\), and then determine the value of \(k\) from the given conditions. \(u\) is directly proportional to \(v\). If \(v=30\), then \(u=12\).

Step-by-Step Solution

Answer

The formula is \(u = \frac{2}{5} v\) with \(k = \frac{2}{5}\).

Step 1: Understand Direct Proportionality

When a quantity \(u\) is directly proportional to another quantity \(v\), it means that \(u\) can be expressed as \(u = k \cdot v\). Here, \(k\) is the constant of proportionality.

Step 2: Write the Proportionality Formula

Based on the information given, express the relationship using the formula: \(u = k \cdot v\).

Step 3: Substitute the Given Values

Substitute \(v = 30\) and \(u = 12\) into the formula: \[12 = k \cdot 30\]

Step 4: Solve for \(k\)

Solve the equation \(12 = k \cdot 30\) for \(k\). Divide both sides by 30 to get \[k = \frac{12}{30} = \frac{2}{5}\].

Key Concepts

Constant of Proportionality

The constant of proportionality is a key concept in understanding direct relationships between two quantities. It essentially tells us how much one variable changes when the other variable changes. In the context of direct proportionality, this constant is the fixed multiplier that connects the two variables. For our exercise, the relationship between variables is captured by the equation \( u = k \cdot v \), where \( k \) represents the constant of proportionality.

This constant remains unchanged as long as the relationship between the variables stays direct, meaning they change in a consistent manner. If \( u = 12 \) when \( v = 30 \), solving for \( k \) involves rearranging the equation to \( k = \frac{12}{30} \). Simplifying this gives \( k = \frac{2}{5} \). Thus, \( k \) quantifies how \( u \) scales with \( v \).

A few important things to remember about the constant of proportionality:

• It is specific to each proportional relationship.
• It must be consistent throughout the relationship.
• Changing either variable independently of this relationship changes the constant.

Understanding the constant of proportionality helps in predicting one variable's behavior as the other variable changes.

Proportional Relationship

A proportional relationship is a relationship between two variables where their ratio is constant. In these relationships, as one quantity increases, the other quantity increases at a consistent rate, and the same applies for decreases.

In our example, the proportional relationship is expressed as \( u = k \cdot v \). This equation shows that the ratio between \( u \) and \( v \) is \( k \), the constant of proportionality.

This type of relationship is linear, meaning a graph of this relationship will be a straight line passing through the origin, showcasing a steady, linear increase or decrease.

Some key aspects of proportional relationships include:

• The graph is a straight line.
• The line passes through the origin (0,0).
• The slope of the line represents the constant \( k \).

Recognizing whether two quantities have a proportional relationship helps in solving various practical problems, allowing for predictions about one quantity when the other changes.

Algebraic Expressions

Algebraic expressions are a combination of numbers, variables, and operations that are used to represent mathematical ideas. They are fundamental in formulating equations that express relationships such as direct proportionality.

In the exercise, the algebraic expression \( u = k \cdot v \) is used to model the relationship between the variables \( u \) and \( v \). Here, \( u \) and \( v \) are variables, \( k \) is a constant, and the multiplication operation connects them.

When dealing with algebraic expressions in proportional relationships:

• Identify the variables involved.
• Determine the operations connecting them (here, multiplication).
• Recognize any constants that dictate the relationship dynamics.

Breaking down algebraic expressions helps in understanding their components and how they relate to each other. This understanding is crucial both in forming the correct proportionality equations and solving them correctly by manipulating the variables and constants involved.