Problem 1 - Thomas Calculus

Question

On a circle of radius 10 \(\mathrm{m}\) , how long is an arc that subtends a central angle of (a) 4\(\pi / 5\) radians? (b) \(110^{\circ} ?\)

Step-by-Step Solution

Answer

(a) Arc length is \(8\pi\) meters. (b) Arc length is \(\frac{55\pi}{9}\) meters.

Step 1: Understanding the Formula for Arc Length

The length of an arc of a circle can be computed using the formula \( L = r \theta \), where \( L \) is the arc length, \( r \) is the radius of the circle, and \( \theta \) is the central angle in radians.

Step 2: Convert Angle from Degrees to Radians (if necessary)

For part (b), we must convert \(110^{\circ}\) to radians. The conversion is done using the formula: \( \theta_{radians} = \theta_{degrees} \times \frac{\pi}{180}\). So, \(110^{\circ} = 110 \times \frac{\pi}{180} = \frac{11\pi}{18}\) radians.

Step 3: Calculate Arc Length for Angle in Radians (Part a)

Use the formula \( L = r \theta \) for the angle provided in radians, \( \theta = \frac{4\pi}{5} \). Thus, \( L = 10 \times \frac{4\pi}{5} = 8\pi \) meters.

Step 4: Calculate Arc Length for Angle in Radians (Part b)

Now that \(110^{\circ}\) is converted to \( \frac{11\pi}{18} \) radians, use the arc length formula. Thus, \( L = 10 \times \frac{11\pi}{18} = \frac{110\pi}{18} \) meters, which simplifies to \( \frac{55\pi}{9} \) meters.

Key Concepts

Central Angle

A central angle is the angle formed at the center of a circle between two radii. In a circle, the central angle helps us determine many other aspects like the arc length or the area of a sector. The arc length that the central angle subtends depends directly on the size of this angle and the radius of the circle.
The formula for calculating the arc length when you know the central angle in radians is simple:

• Arc Length Formula: \( L = r \theta \)

Where:

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

This direct relationship means if the central angle doubles, so does the arc length! When working on problems, it is important to ensure the central angle is always in radians while applying this formula. This will ensure accurate calculations and understanding.

Radians to Degrees Conversion

Often in math and science, we need to convert angles from degrees to radians or vice versa. This conversion is necessary because different formulas and applications may require the angle in a specific unit. Degrees are commonly used in everyday situations, while radians are the standard unit in mathematical contexts.
To convert degrees to radians, use the following conversion formula:

• \( \theta_{radians} = \theta_{degrees} \times \frac{\pi}{180} \)

For example, if you have 110 degrees and want to convert it to radians, calculate:

• \( 110^{\circ} \times \frac{\pi}{180} = \frac{11\pi}{18} \) radians

This conversion process is vital when calculating arc lengths or dealing with other circle-related calculations where radians are required. Remember, converting accurately helps maintain the integrity of your results.

Circle Geometry

Circle geometry is a branch of geometry focused on the properties and relationships of points, lines, angles, and figures created on a circle. It includes concepts like arcs, sectors, tangents, and chords, which are all quite interconnected. Understanding circle geometry can be crucial when solving problems involving circles as it lays the foundation for various other calculations.
Key elements in circle geometry include:

• Radius: The distance from the center of the circle to any point on its circumference.
• Diameter: Twice the radius, the longest distance across the circle.
• Circumference: The total distance around the circle, calculated by \( 2\pi r \).
• Arc: A portion of the circle, measured in terms of its length along the circumference.
• Sector: The area enclosed by two radii and an arc.

Solving problems in circle geometry often involves using these elements and understanding their relationships. For instance, knowing the radius and central angle can help calculate an arc's length, like in the provided exercise. Embracing these circle properties enhances both geometry comprehension and practical problem-solving skills.