Mathematics For Machine Technology
8th Edition
ISBN:9781337798310
Author:Peterson, John.
Publisher:Peterson, John.
Chapter44: Solution Of Equations By The Subtraction, Addition, And Division Principles Of Equality
Section: Chapter Questions
Problem 38A
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![### Question 14
**Find the length of arc \( \overarc{AB} \).**
The provided diagram is a circle with a radius of 4 units. The central angle subtended by the arc \( \overarc{AB} \) is 120°. The radius, \( OA \) or \( OB \), is shown as 4 units each.
To find the length of arc \( \overarc{AB} \), you can use the formula for the length of an arc:
\[ L = r \theta \]
Where:
- \( L \) is the length of the arc
- \( r \) is the radius of the circle
- \( \theta \) is the central angle in radians
First, convert the central angle from degrees to radians:
\[ \theta = 120^\circ \times \frac{\pi}{180^\circ} = \frac{2\pi}{3} \, \text{radians} \]
Now, substitute the values into the arc length formula:
\[ L = 4 \times \frac{2\pi}{3} \]
\[ L = \frac{8\pi}{3} \]
Thus, the length of the arc \( \overarc{AB} \) is \(\frac{8\pi}{3}\) units.
**Your answer:**
\[ \frac{8\pi}{3} \text{ units} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd9a023c6-912c-46a5-9561-bfaf13c3fa3b%2F43e2ab2f-0377-4f48-aa97-fa4714b5ca5e%2F8e1fm3n_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Question 14
**Find the length of arc \( \overarc{AB} \).**
The provided diagram is a circle with a radius of 4 units. The central angle subtended by the arc \( \overarc{AB} \) is 120°. The radius, \( OA \) or \( OB \), is shown as 4 units each.
To find the length of arc \( \overarc{AB} \), you can use the formula for the length of an arc:
\[ L = r \theta \]
Where:
- \( L \) is the length of the arc
- \( r \) is the radius of the circle
- \( \theta \) is the central angle in radians
First, convert the central angle from degrees to radians:
\[ \theta = 120^\circ \times \frac{\pi}{180^\circ} = \frac{2\pi}{3} \, \text{radians} \]
Now, substitute the values into the arc length formula:
\[ L = 4 \times \frac{2\pi}{3} \]
\[ L = \frac{8\pi}{3} \]
Thus, the length of the arc \( \overarc{AB} \) is \(\frac{8\pi}{3}\) units.
**Your answer:**
\[ \frac{8\pi}{3} \text{ units} \]
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