Find the length of the arc on a circle of radius r intercepted by a central angle 0. Round to two decimal places. Use x=3.141593. r= 3 yards, 0=65° OA. 3.06 yards O B. 3.74 yards O C. 3.40 yards O D. 2.72 yards
Angles in Circles
Angles within a circle are feasible to create with the help of different properties of the circle such as radii, tangents, and chords. The radius is the distance from the center of the circle to the circumference of the circle. A tangent is a line made perpendicular to the radius through its endpoint placed on the circle as well as the line drawn at right angles to a tangent across the point of contact when the circle passes through the center of the circle. The chord is a line segment with its endpoints on the circle. A secant line or secant is the infinite extension of the chord.
Arcs in Circles
A circular arc is the arc of a circle formed by two distinct points. It is a section or segment of the circumference of a circle. A straight line passing through the center connecting the two distinct ends of the arc is termed a semi-circular arc.
![### Arc Length Calculation
#### Problem Statement:
Find the length of the arc on a circle of radius \( r \) intercepted by a central angle \( \theta \). Round to two decimal places. Use \( \pi = 3.141593 \).
#### Given:
- Radius (\( r \)): 3 yards
- Central angle (\( \theta \)): 65 degrees
### Possible Answers
A. 3.06 yards\
B. 3.74 yards\
C. 3.40 yards\
D. 2.72 yards
### Solution:
To find the arc length (\( L \)), use the formula:
\[ L = r \cdot \theta \cdot \left(\frac{\pi}{180}\right) \]
Where:
- \( r \) is the radius of the circle.
- \( \theta \) is the central angle in degrees.
- \( \pi \) is a constant (approximately 3.141593).
### Detailed Calculation:
1. Convert the central angle from degrees to radians:
\[ \theta = 65^\circ \]
\[ \theta \text{ (in radians)} = 65 \times \left(\frac{\pi}{180}\right) \]
2. Substitute the values:
\[ L = 3 \cdot 65 \cdot \left(\frac{3.141593}{180}\right) \]
3. Simplify the expression:
\[ L = 3 \cdot 65 \cdot 0.01745329252 \]
\[ L = 3 \cdot 1.134115014 \]
\[ L \approx 3.402345042 \]
4. Round to two decimal places:
\[ L \approx 3.40 \text{ yards} \]
### Answer:
C. 3.40 yards
### Interactive Component:
Click to select your answer:
- \( \circ \ \) **A. 3.06 yards**
- \( \circ \ \) **B. 3.74 yards**
- \( \circ \ \) **C. 3.40 yards**
- \( \circ \ \) **D. 2.72 yards**
### Explanation:
This solution demonstrates the process of calculating the length of an arc intercepted by a central angle in a circle, using the given radius and angle measurement in](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fad727bdc-baba-4947-a2e3-e934fc451c32%2F02eb2e49-130a-42de-bf1f-78f9fde1b4be%2Fojwpsk_processed.jpeg&w=3840&q=75)
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