Problem 54 - Algebra A Combined Function

Question

Write an equation to describe each variation. Use \(k\) for the constant of proportionality. \(y\) varies directly as \(a^{5}\) and inversely as \(b\)

Step-by-Step Solution

Answer

The equation is \( y = \frac{k \cdot a^5}{b} \).

Step 1: Understand Direct Variation

When a variable varies directly as another variable, it means that the two variables increase or decrease together at the same rate. Mathematically, if \( y \) varies directly as \( a^5 \), it can be described by the equation \( y = k \cdot a^5 \), where \( k \) is the constant of proportionality.

Step 2: Understand Inverse Variation

When a variable varies inversely as another variable, it means that one variable increases while the other decreases. If \( y \) varies inversely as \( b \), it can be described by the equation \( y = \frac{k}{b} \), where \( k \) is the constant of proportionality.

Step 3: Combine Direct and Inverse Variation

We need to combine both types of variations since \( y \) varies directly as \( a^5 \) and inversely as \( b \). We can express the relationship as \( y = \frac{k \cdot a^5}{b} \). This equation correctly represents the given conditions.

Key Concepts

Proportionality Constant

When dealing with variation, the concept of a proportionality constant, often denoted as \(k\), is fundamental. This constant links two variables, showing the strength and direction of their relationship. In mathematics, the proportionality constant helps quantify how much one variable changes as another variable changes.

In equations of direct or inverse variation, \(k\) acts as a fixed multiplier. For example:

• In direct variation, like \( y = k \cdot a^5 \), \(k\) represents how strongly \(y\) changes with \(a^5\).
• In inverse variation, like \( y = \frac{k}{b} \), \(k\) indicates the relationship between \(y\) and \(b\) when \(b\) changes.

Often, \(k\) must be determined through observations or calculations, allowing you to understand the specific dynamics of the variables involved. This constant helps in creating predictions and understanding relationships within data, emphasizing its significance in real-world problem-solving.

Direct Variation

Direct variation is when two variables move together in the same direction. As one variable increases, the other also increases, and vice versa. Mathematically, if \( y \) varies directly as \( a^5 \), it is expressed as \( y = k \cdot a^5 \). Here, \(a^5\) is raised to the power of 5, but the core idea remains: the variables change together proportionally.

When dealing with direct variation:

• As \(a^5\) increases, \(y\) increases by the same proportion.
• As \(a^5\) decreases, \(y\) decreases similarly.

This direct relationship is often seen in simple systems where changes are predictable and proportional, such as speed and distance or time and work. The capacity to predict outcomes makes direct variation invaluable in scientific calculations and day-to-day applications.

Inverse Variation

Inverse variation occurs when one variable increases as the other decreases. This relationship is opposite to direct variation and is expressed as \( y = \frac{k}{b} \) when \(y\) varies inversely with \(b\). Here, \(k\) is the constant of proportionality, acting as a bridge between \(y\) and \(b\).

The key features of inverse variation include:

• As \(b\) increases, \(y\) decreases.
• As \(b\) decreases, \(y\) increases.

Inverse variation is commonly seen in physics and engineering, such as in the relationship between pressure and volume in gases (Boyle’s Law). This concept helps describe systems where an increase in input results in a decrease in output, offering critical insights in scientific and mathematical analyses.