Problem 29 - Algebra 2 and Trigonometry

Question

In \(28-37, \theta\) is the radian measure of a central angle that intercepts an arc of length \(s\) in a circle with a radius of length \(r .\) If \(\theta=4.5\) and \(s=9,\) find \(r\)

Step-by-Step Solution

Answer

The radius \(r\) is 2.

Step 1: Understanding the Relationship

The formula that relates a central angle in radians (\(\theta\)), arc length (\(s\)), and radius (\(r\)) is given by the equation \(s = r \times \theta\). This equation allows us to find any missing component if the other two are known.

Step 2: Substitute Known Values

We know \(\theta = 4.5\) and \(s = 9\). Substituting these values into the formula, we have: \(9 = r \times 4.5\).

Step 3: Solve for \(r\)

To solve for \(r\), we need to isolate \(r\) on one side of the equation. To do this, divide both sides of the equation by \(4.5\): \(r = \frac{9}{4.5}\).

Step 4: Calculate \(r\)

Perform the division: \(r = \frac{9}{4.5} = 2\). Therefore, the radius \(r\) is 2.

Key Concepts

Radians

When we talk about radians, we are referring to a way of measuring angles using the radius of a circle. If you're familiar with degrees, where a full circle is divided into 360 parts, radians offer another perspective. A complete circle in radians is expressed by the number \(2\pi\), which approximately equals 6.283. This means one radian is like slicing a pizza in such a way that the length of the slice is equal to the radius of that pizza.

To convert between degrees and radians, you can use these handy conversions:

• 1 radian = \(\frac{180}{\pi}\) degrees
• 1 degree = \(\frac{\pi}{180}\) radians

Understanding radians is essential, especially when dealing with the geometry of circles, as it naturally ties in with formulas relating to arc length and other circular measurements.

Arc Length

The arc length is the distance you would travel if you were walking along the curved edge of a part of a circle. Imagine taking a piece of string and laying it along a segment of a circular track. The length of that string is the arc length.

The formula to find the arc length when the central angle is measured in radians is:

• \[ s = r \times \theta \]

Here, \(s\) represents the arc length, \(r\) is the radius of the circle, and \(\theta\) is the angle in radians. This formula is incredibly useful because it directly ties together these three aspects of a circle. If any two values are known, you can easily calculate the third.

Circle Radius

The radius of a circle is the distance from its center to any point on its edge. This simple yet crucial measure offers a way to understand and relate to other properties of the circle. As we've seen in the formula for arc length \(s = r \times \theta\), the radius not only affects the size of the circle, but also plays a key role in calculating the arc length when given an angle.

When problems provide certain values, such as an arc length and a central angle, you can rearrange the arc length formula to solve for the radius:\

• \[ r = \frac{s}{\theta} \]

This formula is helpful when you are trying to find out all about the circle's dimensions given specific parameters. Understanding how to manipulate this equation can empower you to solve a variety of geometry and trigonometry problems, especially those involving circle-related calculations.