Problem 72 - Elementary Algebra

Question

Suppose that the length of a radius of a circle is doubled. How does this affect the area of the circle? Defend your answer.

Step-by-Step Solution

Answer

The area becomes four times larger when the radius is doubled.

Step 1: Understanding the Original Circle's Area

The formula to find the area of a circle is given by \[ A = \pi r^2 \] where \( r \) is the radius of the circle. Review this formula as it is important for calculating the area before and after any change to the radius.

Step 2: Calculating Area of Increased Radius

If the radius is doubled, the new radius is given by \[ r_{new} = 2r \]The new area of the circle with radius doubled is \[ A_{new} = \pi (2r)^2 \] which simplifies to \[ A_{new} = \pi \times 4r^2 = 4\pi r^2 \].

Step 3: Comparing Areas

Compare the new area, \(A_{new} = 4\pi r^2\), to the original area, \(A = \pi r^2\). Notice that \[ A_{new} = 4 \times A \] This shows that the area of the circle becomes four times the original area when the radius is doubled.

Key Concepts

Area of a Circle

One of the fundamental concepts in circle geometry is the area of a circle. When we talk about the area, we mean the amount of space enclosed within the circle's boundary. This is crucial for many applications, such as calculating materials needed for circular designs or finding out how much space an object can occupy.

The formula to determine the area of a circle is given by \[ A = \pi r^2 \]where \( A \) is the area, \( \pi \) is a constant approximately equal to 3.14159, and \( r \) represents the radius of the circle. This formula comes from calculus and geometric reasoning, all grounding it in practical measurement. Understanding this formula is essential for accurately evaluating the space inside any circle. The larger the value of the radius, the more significant the area will be.

Radius

The radius of a circle is a straight line from its center to any point on its boundary. It's an essential feature that determines not only the size but also the various attributes of the circle, like its area and circumference.

Key points to understand about the radius include:

• It is half the length of the diameter—the longest chord through the center of the circle.
• Altering the radius affects the whole shape and its properties due to its quadratic presence in calculations like area and circumference.
• In terms of units, the radius \( r \) is often measured in meters, centimeters, or any unit of length.

Recognizing the role of the radius helps in solving numerous circle-related problems, from assessing space to understanding expansions and contractions in geometric terms.

Impact of Radius on Area

The radius has a profound impact on the area of a circle due to its squared presence in the area formula. This means that any change in the radius will have an exponential effect on the area.

Consider this, when you double the radius of a circle:

• The new radius becomes \( 2r \).
• Plugging this in the area formula results in:\[ A_{new} = \pi (2r)^2 = 4\pi r^2 \].
• This equation shows that when the radius is doubled, the area quadruples, or becomes four times larger.

Such exponential increase highlights the power of the radius in determining the space a circle covers. Consequently, designers, engineers, and mathematicians always account for this when working with circular objects, demonstrating how vital it is to consider changes in radius during calculations.

Mathematical Formulas

Mathematical formulas are tools that help us compute values efficiently and accurately. Formulas like the area of a circle formula allow for systematic and repeatable calculations, essential in various fields such as architecture, engineering, and physics.

To comprehend the relation between radius and area:

• The base formula is \[ A = \pi r^2 \],which means the area directly depends on the square of the radius.
• When adjustments occur to the radius, like doubling it: \[ A_{new} = \pi (2r)^2 \], we can manipulate the formula to find new areas.

These formulas serve as a universal language in mathematics, offering clarity and simplifying complex problems into manageable mathematical equations. This understanding equips you with the knowledge to predict and compute quantifiable changes efficiently.