Problem 13 - Beginning and Intermediate Algebra

Question

Write a general variation equation using \(k\) as the constant of variation. \(T\) varies inversely as the square of \(c\)

Step-by-Step Solution

Answer

The general variation equation for the given relationship is \(T = \frac{k}{c^2}\), where \(T\) varies inversely as the square of \(c\), and \(k\) is the constant of variation.

Step 1: Define the inverse variation relationship

Since T varies inversely as the square of c, we can write this relationship as follows: \(T = \frac{k}{c^2}\) Here, k is the constant of variation.

Step 2: Write the general variation equation

Now, we have the general variation equation involving T, c, and the constant of variation k: \(T = \frac{k}{c^2}\)

Key Concepts

General Variation Equation

When we talk about a general variation equation in algebra, we're discussing how one variable changes concerning another. Variations can be direct or inverse. For instance, in direct variation, if one variable increases, the other also increases. Conversely, inverse variation means if one variable increases, the other decreases.

In the provided exercise, we see a classic example of inverse variation where the variable \( T \) varies inversely as the square of \( c \). The general equation for this kind of variation is given by:

• \( T = \frac{k}{c^2} \)

Here, \( k \) signifies the constant of variation, a key component that remains unchanged as \( T \) and \( c \) fluctuate. This equation illustrates how the product \( T \, c^2 \) is always a constant \( k \). By grasping this concept, you can apply it to various scenarios involving inverse variation.

Constant of Variation

The constant of variation, denoted by \( k \), plays a crucial role in any variation equation. It acts as the fixed value that relates two variables, signifying the relationship between them remains consistent.

In our equation, \( T = \frac{k}{c^2} \), the constant \( k \) allows us to understand how changes in \( c \) affect \( T \). No matter what values \( c \) might take, the product \( T \, c^2 \) will always equal \( k \).

• The constant of variation is essential as it provides a means to calculate unknown quantities given known ones.
• It also tells us something about the initial relationship between the variables \( T \) and \( c \).

Understanding \( k \) helps in appreciating the predictability and order within variable relationships.

Algebraic Relationships

Algebraic relationships involve the connection between variables using equations. These relationships help us describe and predict how one variable behaves as another changes.

In the context of inverse variation, like in our exercise, algebraic relationships are formed by expressing one variable as inversely proportional to another. Algebra operates under universal rules, providing consistency in how equations such as:

• \( T = \frac{k}{c^2} \)

are applied. This allows us to:

• Predict the behavior of \( T \) for different values of \( c \).
• Formulate strategies for similar problems by identifying patterns in variable interactions.

By understanding algebraic relationships, you can solve complex problems by breaking them into manageable equations, painting a clear picture of each variable's role. This clarity is essential in both learning and applying mathematical concepts effectively.