Problem 53 - Algebra and Trigonometry

Question

Find the length of an arc that subtends a central angle of 2 rad in a circle of radius 2 \(\mathrm{mi} .\)

Step-by-Step Solution

Answer

The length of the arc is 4 miles.

Step 1: Understand the Arc Length Formula

The formula to find the length of an arc is given by \( L = \theta \times r \), where \( L \) is the arc length, \( \theta \) is the central angle in radians, and \( r \) is the radius of the circle.

Step 2: Identify Given Values

In the problem, the central angle \( \theta \) is 2 radians, and the radius \( r \) is 2 miles.

Step 3: Substitute Values into the Formula

Substitute the given values into the arc length formula: \( L = 2 \times 2 \).

Step 4: Calculate the Arc Length

Perform the multiplication to find the length of the arc: \( L = 4 \).

Step 5: Conclusion

The length of the arc is 4 miles.

Key Concepts

Central Angle

The central angle is an important concept in the realm of circles. It is essentially the angle subtended at the center of the circle by two radii. In other words, it is the angle formed at the circle's center when two radii extend to the edge of the circle, creating an arc. Understanding the central angle is crucial when dealing with arc length, as it directly influences it.

Here are some key points about the central angle:

• A central angle can be measured in degrees or radians. In many mathematical applications, radians are preferred as they simplify many calculations involving circles.
• The central angle is denoted by the symbol \( \theta \).
• For a full circle, the central angle is \( 2\pi \) radians or 360 degrees.
• The size of the central angle determines the size of the arc: the larger the angle, the larger the arc.

Having a thorough understanding of central angles helps in comprehending the general properties of circles and in aspects of geometry involving arc calculations.

Circle Radius

The radius of a circle is another fundamental component when calculating arc lengths. It is the straight-line distance from the center of the circle to any point on its circumference. The length of the radius plays a significant role in the geometry of a circle.

Key aspects of circle radius include:

• The radius is half the diameter of the circle.
• All radii in a circle are equal in length since every point on the circumference is equidistant from the center.
• The radius is denoted by the letter \( r \).
• For practical calculations, knowing the radius allows one to determine various properties of the circle, including the arc length when combined with the central angle.

Understanding the radius is crucial for any calculation involving the size and properties of the circle, such as finding the circumference, area, and particularly, arc length.

Radian Measure

Radians are a natural way to express angles in mathematics and are often used in calculations involving circles. Unlike degrees, which divide a circle into 360 equal parts, radians provide a measure based on the radius of the circle. This is both logical and practical for mathematical purposes.

Important features of radian measure include:

• One radian is the angle formed when the arc length is equal to the radius of the circle.
• A full rotation around a circle is \( 2\pi \) radians, equivalent to 360 degrees.
• An angle of \( \pi \) radians equals 180 degrees, making \( \pi \) a crucial conversion factor from radians to degrees and vice versa.
• Using radians simplifies the mathematical formulas for arc length \( L = \theta \times r \), making calculations more intuitive.

Mastering radian measure enables smoother and more efficient problem-solving in mathematics, particularly when it comes to angle and arc length calculations in circular geometry.