Problem 82 - Precalculus

Question

Find the exact length of the are made by the indicated central angle and radius of each circle. $$\theta=\frac{\pi}{8}, r=6 \text { yd }$$

Step-by-Step Solution

Answer

The arc length is \( \frac{3\pi}{4} \) yards.

Step 1: Identify the Formula

To find the arc length, we use the formula: \[ L = r \cdot \theta \] where \( L \) is the arc length, \( r \) is the radius, and \( \theta \) is the angle in radians.

Step 2: Substitute the Known Values

We know \( r = 6 \) yards and \( \theta = \frac{\pi}{8} \). Substitute these values into the formula: \[ L = 6 \cdot \frac{\pi}{8} \]

Step 3: Calculate the Arc Length

Simplify the expression to find the arc length: \[ L = \frac{6\pi}{8} = \frac{3\pi}{4} \] Thus, the exact length of the arc is \( \frac{3\pi}{4} \) yards.

Key Concepts

Central Angle

A central angle is an essential concept in understanding circles. It is an angle whose vertex lies at the center of a circle. This angle opens towards the circumference, formed by two radii. Imagine a pizza slice; the tip of the slice represents the central angle. Central angles are significant because they help us determine other geometric properties of the circle, such as arc length and sector area.
Without the knowledge of the central angle, calculating these properties would be challenging. Depending on the measurement unit, central angles can be expressed in degrees or radians. In our problem, the central angle is given in radians, which plays a critical role in further calculations involving arc length. By using radians, we can directly integrate this value into mathematical formulas without conversion.

Radius

The radius is another fundamental concept in the study of circles. It is the distance from the center of a circle to any point on its circumference. In simpler terms, it's like the spoke of a bicycle wheel, extending from the center to the outer edge. The radius is central to calculating various properties of a circle, including the arc length and area. In our specific exercise, the radius is given as 6 yards. This value is directly used in the formula for arc length.

• Radius ( ): Plays a vital role in circle-related calculations.
• It is always a positive value and invariant for a given circle.

Understanding the concept of radius is crucial as it acts as a building block for more complicated geometric calculations.

Radians

Radians are a unique way of measuring angles, distinct from the traditional degrees most people are familiar with. Instead of dividing the circle into 360 degrees, radians measure angles based on the circle's radius.
A full circle is equal to 2π radians.This unit of measurement is essential in mathematics, especially when working with trigonometric functions and calculus, because it relates directly to the intrinsic properties of circles.

• Radians provide a natural measure of angles.
• Helpful for equations involving arc length and sector area.

In our problem, the angle given is \(\theta = \frac{\pi}{8}\) radians, which indicates the section of the circle to consider when calculating arc length. Expressing angles in radians allows for more straightforward integration into circle formulas.