Problem 81 - Precalculus

Question

Find the exact length of the are made by the indicated central angle and radius of each circle. $$\theta=\frac{\pi}{12}, r=8 \mathrm{ft}$$

Step-by-Step Solution

Answer

The exact length of the arc is \(\frac{2\pi}{3}\) feet.

Step 1: Understanding the Formula

To find the arc length of a circle, we use 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: Substituting the Values

Given in the problem, the radius \(r = 8\) ft and the central angle \(\theta = \frac{\pi}{12}\). Substitute these values into the formula:\[L = 8 \times \frac{\pi}{12}\]

Step 3: Simplifying the Expression

Now, simplify the expression step-by-step:\[L = 8 \times \frac{\pi}{12} = \frac{8\pi}{12} = \frac{2\pi}{3}\]

Step 4: Final Calculation

Thus, the length of the arc is \(\frac{2\pi}{3}\). Since \(\pi\) is a known constant (approximately 3.14159), this can be left in terms of \(\pi\) or further calculated if needed. For an exact value, it remains as \(\frac{2\pi}{3} \approx 2.094\).

Key Concepts

Central Angle

In the study of circles, the central angle is an angle whose vertex is located at the center of a circle. This angle is crucial as it helps in calculating the arc length and other geometrical properties of the circle. The central angle is measured in radians or degrees.

• When measured in radians, it directly influences the arc length when multiplied by the radius.
• One complete circle is equal to an angle of \(2\pi\) radians or 360 degrees.

In the given exercise, the central angle is \(\theta = \frac{\pi}{12}\). This indicates that the angle subtends a small part of the circle due to its small value compared to \(2\pi\). Understanding how central angles interact with other circle properties is essential for grasping circle geometry.

Radius

The radius of a circle is the fixed distance from the center of the circle to any point on the boundary. It plays a key role in calculations concerning the circle’s area, circumference, and particularly, its arc length. The radius is often symbolized as \(r\). In our exercise, the radius is given as 8 feet.

• Different radii create different-sized circles, but they all share the property of constant distance from the center to the circle’s edge.
• The larger the radius, the larger the circle and its respective segments like arcs and sectors.

When calculating arc length, you multiply the radius by the central angle measured in radians. Comprehending how the radius scales with these measurements is integral to mastering circle-related problems.

Radians

Radians are a unit of angular measure used in many fields of mathematics. Understanding radians is crucial when working in trigonometry and calculus, particularly with circles. Unlike degrees, radians provide a direct relationship to the circle's radius using the formula for arc length.

• One radian is the angle created when the radius length forms an arc equivalent to the radius along the circle's circumference.
• There are \(2\pi\) radians in a full circle (360 degrees), making it a natural way to measure angles in the context of circles.

In the exercise, the central angle, \(\theta = \frac{\pi}{12}\), is expressed in radians. This expression allows us to directly calculate the arc length by multiplying with the radius, applying the formula \(L = r\theta\). By mastering radians, students can easily translate angular measurements into linear distances along the circle.

Circle

A circle is a two-dimensional shape defined by all points equidistant from a central point known as the center. This equal distance is known as the radius. Circles are fundamental in geometry and have numerous applications in math and real-world contexts.

• A circle’s entire boundary is called the circumference, calculated as \(2\pi r\) where \(r\) is the radius.
• Sections of the circumference are arcs, defined by their start and end points and the circle's central angles.
• The area inside the circle is given by the formula \(\pi r^2\).

Grasping the basic properties of circles, such as how the radius and central angle determine arcs, is fundamental. It allows students to solve various problems involving circle geometry, such as finding arc lengths or areas of circular segments.