Problem 55 - 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 arc length is 4 miles.

Step 1: Understand the Formula for Arc Length

The length of an arc is computed using the formula \( L = r \theta \), where \( L \) is the arc length, \( r \) is the radius of the circle, and \( \theta \) is the central angle in radians.

Step 2: Identify Given Values

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

Step 3: Substitute Values into the Formula

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

Step 4: Perform the Multiplication

Calculate the arc length by multiplying: \( L = 4 \).

Step 5: State the Result

Conclude that the length of the arc is 4 miles.

Key Concepts

Central Angle

The central angle plays a crucial role in determining the length of an arc in a circle. When you look at a circle, the central angle is formed by drawing two radii from the center of the circle to any two points on the circle's edge. This angle essentially 'subtends' or intercepts the arc you're interested in measuring.

• The central angle is expressed in radians for easy geometric calculations.
• For our specific problem, the central angle was given as 2 radians.

Understanding the central angle helps in visualizing what part of the circle the arc encompasses. Imagine cutting a piece of pie - the central angle measures the size of that slice.

Radians

Radians are the standard unit of angular measure used in many areas of mathematics. They provide a natural way to express angles occurring in circular motion or arc measurements. A full circle equates to \(2\pi\) radians, which is approximately 6.28 radians.
Why use radians instead of degrees?

• Radians provide a direct relationship between the radius of the circle and the arc length.
• They make formulas simpler and more consistent when working with angular motion.

In the exercise, the use of radians allowed us to easily apply the formula for arc length, as the central angle was already in a form that worked directly with the circle's radius.

Circle Radius

The radius of a circle is the distance from the center to any point on the circle's edge. It is a key component in finding the arc length. When you know the radius and the measure of the central angle in radians, calculating the arc length becomes straightforward using the formula \( L = r \theta \).

• In our problem, the radius was given as 2 miles.
• The radius is essential because it scales the central angle to determine the actual distance along the circle's edge.

Understanding the role of the radius is essential when dealing with circular figures, whether you're calculating arc lengths or dealing with more complex geometric problems.