Problem 124 - Precalculus

Question

Determine whether each statement is true or false. If the radius of a circle doubles, then the area of the sector (associated with a fixed central angle) doubles.

Step-by-Step Solution

Answer

False. The area of the sector becomes four times larger, not double.

Step 1: Understanding the Area of a Sector

For a circle with radius \( r \) and central angle \( \theta \) (in radians), the area \( A \) of the sector is given by the formula \( A = \frac{1}{2} r^2 \theta \). This formula enables us to determine how the area changes if the radius changes.

Step 2: Initial Area Calculation

If the original radius is \( r \), the area \( A_1 \) of the sector is \( A_1 = \frac{1}{2} r^2 \theta \). This is our starting point for the area calculation with the initial radius.

Step 3: Area with Doubled Radius

If the radius doubles to become \( 2r \), the new area \( A_2 \) of the sector is \( A_2 = \frac{1}{2} (2r)^2 \theta = \frac{1}{2} imes 4r^2 imes \theta \). Simplifying, we get \( A_2 = 2 r^2 \theta \), which is clearly four times \( A_1 \), not double.

Step 4: Conclusion on Statement's Truthfulness

The formula shows that if the radius doubles, the area of the sector becomes four times the original area. Therefore, the statement that the area of the sector doubles is false.

Key Concepts

Circle

A circle is a simple yet fundamental shape in geometry. It consists of all points that are equidistant from a fixed central point, known as the center. This distance is called the radius.
Understanding circles is crucial because they form the basis for more complex shapes and calculations, especially in problems involving curves.
Here are some essential properties of a circle:

• Diameter: The distance across the circle, passing through the center. It is twice the radius.
• Circumference: The perimeter or boundary line of the circle, calculated as \(2\pi r\) where \(r\) is the radius.
• Area: The space enclosed within the circle, calculated using the formula \(\pi r^2\).

Mastery of these properties allows us to solve various geometric problems and understand the behavior of circular shapes better.

Radius

The radius of a circle is a pivotal measurement, playing a crucial role not just in circles but in numerous geometrical and real-world applications.
It is defined as the distance from the center of the circle to any point on the circle's edge.
Here are some key aspects of the radius:

• Proportionality: The radius is directly proportional to several circle properties. For instance, when the radius changes, the area and circumference change accordingly.
• Formulas: Since the radius is central to the calculation of both the circumference (\(2\pi r\)) and area (\(\pi r^2\)), it highlights how essential this measurement is.
• Variability: Changes to the radius can lead to significant differences in other properties, exemplified in scenarios where the radius is doubled, altering areas and perimeters greatly.

Understanding the radius is integral for interpreting how changes to a circle affect its other characteristics.

Sector

A sector is a portion of a circle that resembles a "slice of pie."
It's defined by two radii and the arc between them, creating a specific part of the circle.
In geometry, dealing with sectors is common, and it involves understanding how they relate to the circle's measurement:

• Area Calculation: The area of a sector is calculated using the formula \(\frac{1}{2} r^2 \theta\), where \(r\) is the radius and \(\theta\) is the central angle in radians.
• Central Angle: The angle at the circle's center that defines the sector's 'width'. Its measurement can significantly affect sector properties.
• Radius Influence: As shown in the exercise, changes in the radius can vastly alter the sector's area. Doubling the radius quadruples the area due to the squared factor in the formula.

Understanding sectors is crucial for solving problems involving portions of circles, as they incorporate both the circle's structure and angle measurements into their calculations.