Problem 98 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the length of the arc on a circle of radius \(r\) intercepted by a central angle \(\theta\). Radius \(r\) 9 feet Central Angle \(\theta\) \(\frac{\pi}{3}\) radians

Step-by-Step Solution

Answer

The length of the arc is \(3\pi\) feet.

Step 1: Identify given variables and the formula to use

We are given a circle with radius \(r = 9\) feet and a central angle of \(\theta = \frac{\pi}{3}\) radians. The formula to calculate an arc length is \(arc = r\theta\).

Step 2: Substitute the given values into the formula

If we substitute \(r = 9\) feet and \(\theta = \frac{\pi}{3}\) radians into the formula \(arc = r\theta\), we get \(arc = 9 \cdot \frac{\pi}{3}\).

Step 3: Perform the multiplication to find the arc length

Evaluating the multiplication \(9 \cdot \frac{\pi}{3}\) gives us an arc length of \(3\pi\) feet.

Key Concepts

Central Angle

The central angle of a circle is a critical concept in geometry, especially when dealing with circle-related measurements like arc length. Imagine the center of a circle as a fixed point from which two rays extend outward, creating a wedge-like shape. This shape is what we call the central angle, and it is measured in radians or degrees. In the context of our exercise, the central angle is denoted as \( \theta \), which is \( \frac{\pi}{3} \) radians.

Understanding central angles is important because they determine the portion of the circle’s circumference that forms an arc. Here’s a simple way to think about it:

• The larger the angle, the larger the arc.
• Getting precise measurements requires using the angle's radian or degree value.
• Central angles always originate from the circle's center, making them unique as compared to other angles.

This concept helps in calculating the arc length, which is the distance along the circle's edge that the central angle spans.

Circle Radius

The radius of a circle is the distance from the center of the circle to any point on its perimeter. Symbolized by \( r \), the radius is a fundamental component in various geometric equations and plays a significant role in circle-related problems.

When working with circles, the radius helps define the size of the circle; it directly influences measures like circumference, area, and arc length. Let's focus more on its role in arc length calculation:

• The radius acts as a scaling factor in the arc length formula \( \text{arc length} = r\theta \), where \( \theta \) is the central angle in radians.
• A larger radius means a larger circle, which results in a longer arc for a given central angle.
• In our problem, the radius was given as 9 feet, indicating the circle's size and contributing directly to the resulting arc length calculation.

Knowing the radius helps us understand how dimensions change based on different circle sizes and assists in practical applications like constructing curved paths or designing wheels.

Radian Measure

Radian measure is a system used in mathematics to quantify angles. Unlike degrees, radians offer a natural way to relate angles to the radius of a circle, making them particularly useful in calculus and trigonometry. This system conceptualizes an angle as the length of the arc it intercepts, divided by the radius of the circle.

Here's why radians are significant:

• Radians simplify many formulas and are the Standard Unit of angular measure in mathematics.
• They provide a straightforward approach to defining angles without having to convert between measurement systems.
• One full circle is \( 2\pi \) radians, parallel to 360 degrees in degree measure.

In our specific example, the central angle \( \theta \) is given in radian measure as \( \frac{\pi}{3} \), showing our reliance on this unit for calculating arc length. Using radians simplifies the process and is essential for equations involving circular motion and path length evaluation. Thus, understanding radian measure aids in tackling a variety of geometry problems seamlessly.