Problem 174 - Calculus Volume 1

Question

Find the length of the arc intercepted by central angle \(\theta\) in a circle of radius \(r\) . Round to the nearest hundredth. a. \(r=12.8 \mathrm{cm}, \theta=\frac{5 \pi}{6} \mathrm{rad} \quad \) b. \(r=4.378 \mathrm{cm}, \theta=\frac{7 \pi}{6} \mathrm{rad} \quad\) c. \(r=0.964 \quad \mathrm{cm}, \quad \theta=50^{\circ} \quad\) d. \(r=8.55 \mathrm{cm},\) \(\theta=325^{\circ}\)

Step-by-Step Solution

Answer

Arc lengths: a) 33.51 cm, b) 16.07 cm, c) 0.84 cm, d) 48.51 cm.

Step 1: Understand Arc Length Formula

The formula to find the arc length in a circle is given by \( L = r \cdot \theta \), where \( L \) is the arc length, \( r \) is the radius of the circle, and \( \theta \) is the central angle in radians.

Step 2: Convert Angles to Radians Where Necessary

For parts (c) and (d), where angles are given in degrees, convert them to radians. Use the formula: \( \theta = \theta_{\text{degrees}} \times \frac{\pi}{180} \). Thus, for part (c), \(\theta = 50^{\circ} \approx 0.8727 \) radians. For part (d), \( \theta = 325^{\circ} \approx 5.6723 \) radians.

Step 3: Calculate Arc Length for Each Part

Use the arc length formula and perform the calculations for each given set of \( r \) and \( \theta \):- (a) \( L = 12.8 \times \frac{5\pi}{6} \approx 33.51 \) cm.- (b) \( L = 4.378 \times \frac{7\pi}{6} \approx 16.07 \) cm.- (c) \( L = 0.964 \times 0.8727 \approx 0.84 \) cm.- (d) \( L = 8.55 \times 5.6723 \approx 48.51 \) cm.

Key Concepts

Central Angle

In circle geometry, a central angle is one that has its vertex at the center of the circle. The arms of this angle extend to intersect the circumference, creating an arc. The measure of this central angle is essential when finding the arc's length. It determines how large a portion of the circle the arc covers.

The central angle is always measured in radians or degrees. When using it for calculations involving the arc length, it's important to use radians because the arc length formula directly uses the central angle in radians. Converting from degrees to radians, if necessary, is a crucial step. By understanding the central angle's role, you can easily relate the circle's radius to its arc length.

Radians Conversion

Radians are a unit of measuring angles, often preferred in mathematical calculations involving circles. They provide a straightforward relationship between the radius, central angle, and arc length. A full circle is equal to 2\pi radians, which corresponds to 360 degrees.

To convert an angle from degrees to radians, multiply by \frac{\pi}{180}. For instance:

• Convert 50° to radians: \( 50^{\circ} \times \frac{\pi}{180} \approx 0.8727 \) radians.
• Convert 325° to radians: \( 325^{\circ} \times \frac{\pi}{180} \approx 5.6723 \) radians.

This conversion is vital when using the arc length formula because the formula relies on the angle being in radians. Always remember, for accurate arc length calculations, confirm that the angle is in radians.

Circle Geometry

Circle geometry involves understanding the properties and relationships within a circle, including its center, radius, diameter, circumference, and sectors. These concepts are foundational for solving problems related to circles. A central feature of circle geometry is the relationship between different parts, like the central angle and the arc it intercepts.

The radius is a constant distance from the circle's center to any point on its circumference. The larger the radius, the longer the potential arc length for any given central angle. Understanding how these elements interrelate is crucial for solving problems in circle geometry, particularly when dealing with arc lengths and central angles.

Arc Length Formula

The arc length formula is essential for calculating the portion of the circle's circumference subtended by a central angle. The formula is: \[ L = r \cdot \theta \]where:

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

This formula shows the direct proportional relationship between the arc length and both the radius and the angle. For example, increasing the radius or the angle will increase the arc length. Using this formula, you can effectively find the arc length for any circle if you have the radius and the central angle in radians. Remember, whenever you calculate arc length, the angle must be in radians to ensure accurate results.