Problem 30 - Precalculus: Functions and Graphs

Question

Exer \(29-30:\) If a circular are of the given length \(s\) subtends the central angle \(\theta\) on a circle, find the radius of the circle. $$s=3 \mathrm{km}, \quad \theta=20^{\circ}$$

Step-by-Step Solution

Answer

The radius is \( \frac{27}{\pi} \) km.

Step 1: Understanding the Relationship

The length of an arc, \(s\), in a circle is related to its radius, \(r\), and the central angle, \(\theta\), through the formula: \[s = r \theta\] where \(\theta\) is in radians. First, ensure the angle is in radians.

Step 2: Convert Angle to Radians

To convert degrees to radians, use the formula: \[\theta_{\text{rad}} = \theta_{\text{deg}} \times \frac{\pi}{180}\] Plug in \(\theta = 20^{\circ}\):\[\theta_{\text{rad}} = 20 \times \frac{\pi}{180} = \frac{\pi}{9} \text{ radians}\].

Step 3: Plug Into the Arc Length Formula

Now, substitute \(s = 3\) km and \(\theta_{\text{rad}} = \frac{\pi}{9}\) into the arc length formula: \[3 = r \times \frac{\pi}{9} \].

Step 4: Solve for the Radius

Rearrange the formula to solve for \(r\): \[r = \frac{3 \times 9}{\pi} \]Simplify: \[ r = \frac{27}{\pi} \text{ km} \].

Key Concepts

Radians Conversion

Converting angles from degrees to radians is an essential step when working with circular measurements. Radians are a more natural unit for angular measurements in mathematics because they are directly related to the properties of the circle itself.
To convert from degrees to radians, use the conversion factor \( \frac{\pi}{180} \). This factor comes from the circumference of a circle, where \( 360^{\circ} \) corresponds to \( 2\pi \) radians. Here's how to perform the conversion:

• Identify the angle in degrees. For example, in our problem, \( \theta = 20^{\circ} \).
• Multiply the degree measurement by \( \frac{\pi}{180} \).
• Solve to find the angle in radians: \( \theta_{\text{rad}} = 20 \times \frac{\pi}{180} = \frac{\pi}{9} \).

This conversion is crucial because formulas involving arc length in circles often require angles to be in radians, not degrees.

Central Angle

The central angle is an angle whose vertex is at the center of a circle, and whose sides intersect the circle creating the arc. It's a key concept for understanding relationships in circular geometry.
A central angle can be represented in degrees or in radians. In many applications, including arc length calculations, radians are preferred.
The formula linking the arc length \( s \), the radius \( r \), and the central angle \( \theta \) is:

• \( s = r \theta \)

This tells us that the arc length is proportionate to both the radius and the central angle. Therefore, increasing the central angle while keeping the radius fixed will increase the arc length, and vice versa. Understanding this relationship is vital for solving problems related to the geometry of circles.

Circle Radius

The radius of a circle is the distance from its center to any point on its circumference. It is one of the most fundamental properties of a circle. When dealing with arc length and central angles, the radius becomes a crucial piece in the puzzle.
To find the radius of a circle from a known arc length and central angle, you can rearrange the formula \( s = r \theta \) to solve for \( r \), which gives:

• \( r = \frac{s}{\theta} \)

In the given problem, with \( s = 3 \) km and \( \theta = \frac{\pi}{9} \) radians, we calculate the radius:

• Substitute the values into the formula: \( r = \frac{3}{\frac{\pi}{9}} \)
• Simplify: \( r = \frac{3 \times 9}{\pi} = \frac{27}{\pi} \) km

This method demonstrates how even partial information about a circle's segment can allow us to deduce other important properties.