Problem 97 - A Graphical Approach to Precalculus with Limits

Question

Find the length of each arc intercepted by a central angle \(\boldsymbol{\theta}\) in a circle of radius \(\kappa\) Round answers to the nearest hundredth. $$r=12.3 \text { centimeters; } \theta=\frac{2 \pi}{3} \text { radians }$$

Step-by-Step Solution

Answer

The arc length is 25.75 cm.

Step 1: Understand the Formula for Arc Length

To find the length of an arc (\(s\)) in a circle, use the formula \(s = r\theta\), where \(r\) is the radius of the circle and \(\theta\) is the central angle in radians.

Step 2: Substitute Given Values

Insert the given values into the formula: \(r = 12.3\) cm and \(\theta = \frac{2\pi}{3}\) radians. Thus, the equation becomes \(s = 12.3 \times \frac{2\pi}{3}\).

Step 3: Calculate the Product

Calculate the product by multiplying the values: \(12.3 \times \frac{2\pi}{3} = 12.3 \times 2.094395\) (\(\frac{2\pi}{3} \approx 2.094395\)).

Step 4: Compute the Arc Length

Compute the length: \(s = 12.3 \times 2.094395 = 25.749 \) cm.

Step 5: Round to the Nearest Hundredth

Round the calculated arc length to the nearest hundredth: 25.749 cm rounds to 25.75 cm.

Key Concepts

Understanding the Central Angle

In a circle, the central angle is the angle at the center of the circle, created by two radii extending out to the circumference. Imagine it like a slice of a pizza where the tip of the slice touches the center. This angle helps determine the arc length—a part of the circle's circumference—that it intercepts. It is important to express the central angle in radians rather than degrees when calculating arc length using the formula. This is because radians provide a natural way to describe angles in terms of the radius.

• The formula for the arc length is: \( s = r \theta \)
• This is where \( r \) is the radius, \( \theta \) is the central angle in radians.

The Significance of the Radius

The radius of a circle is the distance from its center to any point on its circumference. Think of it as the line that connects the dot in the center to the edge. The radius is half of the circle's diameter and plays a crucial role in determining the size of the circle. In the context of finding arc length, the radius serves as a scaling factor, multiplying with the central angle (in radians) to calculate the arc length.

• In our exercise, the radius \( r \) is given as 12.3 cm.
• The larger the radius, the larger the potential arc length for the same central angle.

Why Use Radians?

Radians are an essential concept in trigonometry and geometry, often preferred over degrees when dealing with arcs and circles. Unlike degrees, which divide a circle into 360 parts, radians are based on the circle's radius, making them more suitable for abstract mathematical analysis.
Radians relate the angle's measurement directly to the circle's radius and circumference. This makes the radian a naturally efficient measure because the circumference of a circle is \( 2\pi \) times its radius.

• There are \( 2\pi \) radians in a full circle.
• Using radians simplifies calculations involving circles.
• For example, the given central angle is \( \theta = \frac{2\pi}{3} \) radians, not degrees.