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Chapter 9: Problem 16

Describe the combined variation that is modeled by each formula. $$ A=\pi r^{2} $$

Short Answer

Expert verified
The given formula \( A=\pi r^{2} \) represents a direct variation where the area \(A\) of a circle varies directly as the square of its radius \(r\).

Step by step solution

01

Identify the type of variation

Firstly, recognize the type of variation shown in the formula. A variation is said to be direct if the ratio of the two variables is constant. Given the formula \( A=\pi r^{2} \), it can be observed that the area of a circle \(A\) varies directly as the square of the radius \(r\). As such, this represents direct variation.
02

Explain the variation and relationship

Next, explain the relationship between the radius and the area. Since \(A\) varies directly as the square of \(r\), increasing or decreasing the radius will have a direct effect on the area. Namely, if the radius is doubled, the area would quadruple, if the radius is tripled, the area would increase ninefold and so on. This is due to the square in the formula \( r^{2} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Area of a Circle
The area of a circle helps us understand how much space a circle covers on a flat surface. This area is determined using a special formula: \(A = \pi r^2\). In this formula, \(A\) stands for area, \(\pi\) is a mathematical constant approximately equal to 3.14159, and \(r\) is the radius of the circle.

The radius is the distance from the center of the circle to any point on its edge. To find the area, you square the radius (multiply it by itself) and then multiply that result by \(\pi\). This relationship ensures that as the radius changes, so does the area encompassed by the circle.
Relationship Between Radius and Area
The relationship between the radius and the area of a circle is a perfect example of how two variables are related in geometry. The formula \(A = \pi r^2\) tells us that the area of a circle is directly proportional to the square of its radius.

Here is what that means practically:
  • When you double the radius, you do not merely double the area; instead, the area becomes four times larger.
  • If you triple the radius, the area increases by a factor of nine.
  • Similarly, halving the radius will shrink the area to one-fourth of its original size.
This squared relationship emphasizes how small changes in the radius can lead to significant changes in the area of a circle.
Variation Formula
A variation formula like \(A = \pi r^2\) is used to describe how one variable changes with respect to another. This circle area formula represents direct variation. Direct variation implies that the area \(A\) increases as the square of the radius \(r\) increases.

In direct variation:
  • The constant in the equation is \(\pi\), which ties the radius and area together.
  • The constant ratio between area and the square of the radius remains \(\pi\).
Therefore, the area of the circle and the square of the radius move together at this constant rate, maintaining a consistent relationship as radius varies. Direct variation highlights the interconnectedness of these two circle-related measurements.

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