Problem 58 - Algebra and Trigonometry

Question

A circular arc of length 3 \(\mathrm{ft}\) subtends a central angle of \(25^{\circ}\) . Find the radius of the circle.

Step-by-Step Solution

Answer

The radius is approximately 6.897 ft.

Step 1: Understanding the Relationship Between Arc Length and Radius

The length of an arc (\(s\) ) of a circle is directly related to the radius (\(r\) ) and the central angle in radians (\(\theta\) ). The formula for the arc length is: \[ s = r \theta \]

Step 2: Converting Degrees to Radians

Since the given angle is in degrees, we first need to convert it to radians because the arc length formula uses radians. The conversion factor from degrees to radians is \( \frac{\pi}{180}\). Thus, \[ \theta = 25^{\circ} \times \frac{\pi}{180} = \frac{25\pi}{180} = \frac{5\pi}{36} \text{ radians} \]

Step 3: Substituting the Known Values into the Formula

Now that we have \(\theta\) in radians, we can substitute the known values into the arc length formula \( s = r \theta \). We know \( s = 3 \) ft and \( \theta = \frac{5\pi}{36} \). By substituting, we get: \[ 3 = r \times \frac{5\pi}{36} \]

Step 4: Solving for the Radius

To find \(r\), we rearrange the equation: \[ r = \frac{3 \times 36}{5\pi} \] Simplifying, we have: \[ r = \frac{108}{5\pi} \approx 6.897 \text{ ft} \]

Step 5: Conclusion and Verification

So the radius of the circle is approximately 6.897 ft. To verify, calculating back we multiply \(6.897\) by \(\frac{5\pi}{36}\) should give us approximately 3 ft, confirming our solution.

Key Concepts

Arc Length Formula

The arc length formula is a crucial tool for solving circular arc problems. It provides a way to relate the distance along a curve to the circle's radius and the central angle subtended by the arc.
The formula for the arc length, often denoted as \( s \), is given by:
\[ s = r \theta \] , where:

• \( s \) is the arc length,
• \( r \) is the radius of the circle, and
• \( \theta \) is the central angle, measured in radians.

The beauty of this formula lies in its simplicity and how it consistently relates these three quantities. Understanding this formula allows you to find any one of these three values if you have the other two.
When applying this formula, always remember that the central angle must be in radians, not degrees. This ensures the linear relationship between the arc length and the radius remains consistent.
In practical problems, like the exercise we solved, knowing this relationship helped us determine the radius when given the arc length and central angle.

Radians Conversion

Radians are a standard unit of angle measurement in mathematics, especially in trigonometry and calculus. Unlike degrees, which divide a circle into 360 parts, radians measure angles based on the circle's radius.
A full circle is \(2\pi\) radians, which is equivalent to 360 degrees. This makes the conversion between degrees and radians straightforward:

• To convert degrees to radians, multiply by \( \frac{\pi}{180} \).
• For radians to degrees, multiply by \( \frac{180}{\pi} \).

This conversion is crucial, as formulas in mathematics typically use radians. For example, in our problem, we had to convert the 25-degree angle to radians before solving it using the arc length formula.
Understanding how to switch between these units comfortably is essential for solving many mathematical and physics problems. Not only does it ensure accuracy, but it also broadens your understanding of how angles are measured.

Central Angle

The central angle is a key component of circle-related problems. It is the angle that
forms at the center of a circle, with its sides extending to two points on the circumference. This angle plays a crucial role in determining the arc length of a segment of a circle.
A strong grasp of the central angle and its properties allows you to relate various circle elements, such as:

• Arc length, through the formula \( s = r \theta \),
• Sector area, and
• Angle conversions between radians and degrees.

In the exercise example, the central angle was initially given in degrees. Recognizing the need to convert it to radians enabled us to use the arc length formula efficiently.
Overall, the central angle is not just a mere number; it represents a foundational concept that connects different measurements in circular geometry. Mastering this concept will substantially aid in tackling various real-life and theoretical mathematical challenges involving circles.