Problem 8 - Precalculus

Question

Find the measure (in radians) of a central angle \(\theta\) that intercepts an are of length \(s\) on a circle with radius \(r\). \(r=6\) in., \(s=1\) in.

Step-by-Step Solution

Answer

\(\theta = \frac{1}{6}\) radians.

Step 1: Understanding the relationship

The formula relating the arc length \(s\), the radius \(r\), and the central angle \(\theta\) in radians is \(s = r\theta\). We need to find the angle \(\theta\) that satisfies this equation given the values of \(r\) and \(s\).

Step 2: Applying the given values

Use the formula \(s = r\theta\), substituting \(r = 6\) inches and \(s = 1\) inch. This gives \(1 = 6\theta\).

Step 3: Solving for \(\theta\)

Rearrange the equation \(1 = 6\theta\) to solve for \(\theta\). This can be done by dividing both sides of the equation by 6, giving \(\theta = \frac{1}{6}\).

Step 4: Conclusion

The angle \(\theta\) that intercepts an arc of length 1 inch on a circle of radius 6 inches, measures \(\frac{1}{6}\) radians.

Key Concepts

Arc Length

In the realm of circles, arc length is a fascinating concept. It's essentially the distance between two points along the outer edge of the circle. Imagine the arc length as a section of the circle's circumference.
If you ever tried to walk around the edge of a circular garden to measure how much ground you've covered, you were essentially measuring an arc length. For any given arc of a circle, the length depends on two main factors:

• The radius of the circle (how big the circle is)
• The central angle in radians (how wide the angle is that's cutting out the arc)

So, the larger the radius or the angle, the longer the arc. In practical terms, this means the arc length is bigger when you're dealing with a larger slice of the circle or when the circle itself is more enormous in size.

Circle Radius

The circle radius is a crucial part of understanding circles. It's the distance from the center of the circle to any point on its boundary. If you think of the circle as a pizza, the radius would be like the length of a cut made from the center to the crust.
The radius plays an essential role in various circle measurements, like diameter, circumference, and of course, the arc length. The equation that ties these concepts together is all about relationships, such as:

• Diameter being twice the radius, i.e., \(diameter = 2 imes radius\)
• The circumference is calculated using \(Circumference = 2\pi radius\)

The radius is especially pivotal when calculating the arc length with the formula \(s = r\theta\). Here, you multiply the radius by the central angle in radians to find out the arc length. It paints a picture of how an increase in the radius dials up the arc length, even for the same angle.

Central Angle Formula

The central angle formula is a straightforward relation that connects arc length, radius, and the angle. It tells us how big an angle is in terms of radians if you know the arc length and the circle's radius. The formula itself is:\[\theta = \frac{s}{r}\]where:

• \(\theta\) stands for the central angle in radians
• \(s\) is the arc length
• \(r\) is the radius of the circle

This is vital because it allows you to calculate one of these values if you know the other two. In our original problem, by knowing \(s = 1\) inch and \(r = 6\) inches, the central angle \(\theta\) is calculated quickly with the formula:\[\theta = \frac{1}{6}\] radians.
This simplification highlights how the components interact: a smaller arc on a larger radius gives a smaller angle, showcasing the interplay of these geometrical elements.