Problem 63 - Algebra 2

Question

Describe how the variables \(A\) and \(r\) vary in the formula for the area of a circle, \(A=\pi r^{2} .\)

Step-by-Step Solution

Answer

The variables \(A\) and \(r\) vary together - a change in the radius \(r\) causes a change in the area \(A\) of the circle. The relationship is such that if the radius is doubled, the area becomes four times larger, and if the radius is halved, the area becomes a fourth of its original size. This is due to the area being equal to pi times the radius squared, \(A=\pi r^{2}\)

Step 1: Understanding the Formula

The formula \(A= \pi r^{2}\) is used for calculating the area of a circle. Here, \(A\) is the area, \(r\) is the radius of the circle, and \(\pi\) is a constant approximately equal to 3.14.

Step 2: Variation of Radius \(r\)

When the radius \(r\) of the circle increases or decreases, the area \(A\) of the circle also increases or decreases, respectively. This is because the radius \(r\) is directly present in the formula. However, the change is not linear due to the square of the radius \(r^{2}\). So, if the radius doubles, the area will become four times greater. Similarly, if the radius is halved, the area will become a fourth of its original size.

Step 3: Variation of Area \(A\)

The area \(A\) of the circle depends on the square of the radius \(r^{2}\). So, while \(r\) can take any positive value, the minimum value for \(A\) will always be zero when \(r = 0\). As \(r\) increases, \(A\) will increase with the square of \(r\). Therefore, the variation in \(A\) is completely dependent on the changes in \(r\).

Key Concepts

Radius of a Circle

The radius of a circle is a fundamental measurement that spans from the center of the circle to any point on its circumference. This distance is crucial when determining many properties of the circle, such as its area. In the formula for the area of a circle, given by \( A = \pi r^2 \), the variable \( r \) represents the radius. It is essential to understand that the radius should always be a positive number since it represents a physical distance.

When you manipulate the radius, it has a direct and significant impact on the area of the circle. Here's why:

• Doubling the radius means squaring \( 2r \), resulting in four times the original area.
• Conversely, halving the radius results in a quarter of the original area.

This squared relationship ensures that even small changes in the radius can greatly influence the overall area of the circle.

Variation of Parameters

In mathematics, the concept of variation of parameters is often used to study how changes in one parameter affect another. In the context of a circle's area, the variation primarily involves the radius \( r \) and the area \( A \). Understanding this variation can help in predicting how different values of \( r \) affect \( A \).

Consider the formula \( A = \pi r^2 \). As the radius changes, we notice:

• The area increases with the square of the radius, leading to exponential growth.
• For example, an increase in \( r \) from 1 unit to 2 units quadruples the area.

This exponential growth signifies a non-linear relationship, meaning equal changes in the radius do not produce equal changes in the area. This highlights that the area is much more sensitive to variations in the radius than a linear relationship would suggest.

Mathematical Formula Interpretation

Interpreting mathematical formulas involves understanding not just the calculation process but also the relationships and principles they reveal. The formula \( A = \pi r^2 \) exemplifies this vividly:

• \( A \), the area, is proportional to the square of the radius, \( r^2 \).
• The constant \( \pi \), approximately 3.14, is critical as it stabilizes the relationship between \( A \) and \( r^2 \).
• This formula highlights how the geometry of circles cooperates in mathematical principles to define relationships and variations.

Understanding this formula allows for insights into why and how increasing a circle's radius affects its area so dramatically. Recognizing that small changes in the radius lead to large changes in the area helps to grasp the formula's practical applications, such as designing circular objects where space is a concern.