Problem 98 - 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. $$r=0.892 \text { centimeter; } \theta=\frac{11 \pi}{10} \text { radians }$$

Step-by-Step Solution

Answer

The length of the arc is approximately 3.08 cm.

Step 1: Identify Key Formulas and Values

The formula to calculate the arc length \( L \) of a circle intercepted by a central angle \( \theta \) is \( L = \theta \cdot r \). We have \( r = 0.892 \) cm and \( \theta = \frac{11\pi}{10} \) radians.

Step 2: Substitute Values into the Formula

Substitute \( r = 0.892 \) cm and \( \theta = \frac{11\pi}{10} \) radians into the arc length formula: \( L = \frac{11\pi}{10} \times 0.892 \).

Step 3: Calculate the Exact Arc Length

Calculate \( L \) by performing the multiplication: \( L = \frac{11\pi}{10} \cdot 0.892 = 0.892 \cdot \frac{11\pi}{10} = 0.892 \times 3.455 \approx 3.08296 \text{ cm} \) (intermediate calculation before rounding).

Step 4: Round the Arc Length

Round \( L = 3.08296 \text{ cm} \) to the nearest hundredth to get \( L \approx 3.08 \text{ cm} \).

Key Concepts

Central Angle

The concept of a central angle is essential to understand when working with circle geometry. A central angle is formed when two radii extend from the center of the circle to its perimeter. The angle they form at the center is what we refer to as the central angle.
This angle plays a pivotal role in determining the arc length, which is the distance along the circle's edge. The larger the angle, the more of the circle's boundary it covers.
A useful visualization is to imagine a pie. The central angle is like the slice size; a bigger slice means a larger angle.
Understanding and identifying the central angle is crucial when solving problems related to arc lengths.

Radians

When measuring angles, you might be more familiar with degrees, like 90° for a right angle. However, radians are another way to measure angles, especially in mathematics involving circular motion or geometry.
Radians measure angles based on the radius of the circle. A full circle is equal to 2π radians, which is approximately 6.283. Therefore, one radian is the angle formed when the arc length is equal to the circle's radius.
There's a simple relationship between degrees and radians, where 180° is equal to π radians. This makes conversion straightforward: If you have an angle in degrees, you multiply by π/180, and if it's in radians, you multiply by 180/π.
Radians are more natural and useful in math because many formulas, like those in calculus, are simpler using this unit.

Circle Radius

The circle's radius is a fundamental part of many circle-related calculations. It is the distance from the center of the circle to any point along its edge. Not only is it intuitive to imagine, like the arm of a clock, but it also serves as a critical component in numerous formulas.
For instance, when calculating the arc length, the radius is directly multiplied by the central angle to find the arc's measurement. This formula, arc length = radius × central angle, shows the simple yet profound role the radius plays.
A change in the radius will affect the size of the circle, and hence, the arc length for any given central angle. In most practical applications, like the one in this exercise, the radius is provided, allowing you to focus on other aspects of the calculation.