Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
Related questions
Question
![The formula for the length of an arc is given by:
\[ l = \frac{m}{360} \cdot C \]
where \( l \) is the arc length, \( C \) is the circumference, and \( m \) is the measure of the arc in degrees.
The arc's length depends on the arc's measure and the circle's circumference. You already know that the arc's measure is 135°, so find the circle's circumference.
**First, find the circumference:**
\[ C = 2\pi r \]
\[ = 2\pi (3) \quad \text{Plug in } r = 3 \]
\[ = 6\pi \quad \text{Multiply} \]
The circumference is \( 6\pi \) centimeters.
**Now, find the length of the arc:**
\[ l = C \cdot \frac{m}{360} \]
\[ = 6\pi \cdot \frac{135}{360} \quad \text{Plug in } C = 6\pi \text{ and } m = 135 \]
\[ = \frac{9\pi}{4} \quad \text{Multiply and simplify} \]
The length of the arc is \( \frac{9\pi}{4} \) centimeters.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3e360c08-73c4-42e6-a160-35420a0e2834%2F138deef0-3074-4c94-908c-84aa8660c787%2F1lj9bsi_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The formula for the length of an arc is given by:
\[ l = \frac{m}{360} \cdot C \]
where \( l \) is the arc length, \( C \) is the circumference, and \( m \) is the measure of the arc in degrees.
The arc's length depends on the arc's measure and the circle's circumference. You already know that the arc's measure is 135°, so find the circle's circumference.
**First, find the circumference:**
\[ C = 2\pi r \]
\[ = 2\pi (3) \quad \text{Plug in } r = 3 \]
\[ = 6\pi \quad \text{Multiply} \]
The circumference is \( 6\pi \) centimeters.
**Now, find the length of the arc:**
\[ l = C \cdot \frac{m}{360} \]
\[ = 6\pi \cdot \frac{135}{360} \quad \text{Plug in } C = 6\pi \text{ and } m = 135 \]
\[ = \frac{9\pi}{4} \quad \text{Multiply and simplify} \]
The length of the arc is \( \frac{9\pi}{4} \) centimeters.
![**Understanding Arc Length**
**Problem Statement:**
The radius of a circle is 3 miles. What is the length of a 180° arc?
**Diagram Explanation:**
The provided diagram represents a circle with a radius (r) of 3 miles. The diagram highlights a 180° arc, which is half of the circle's circumference.
**Steps to Solve:**
1. **Identify the formula for the circumference of a circle:**
\[
\text{Circumference} = 2 \pi r
\]
2. **Substitute the given radius into the formula:**
\[
\text{Circumference} = 2 \pi \times 3 \text{ miles} = 6 \pi \text{ miles}
\]
3. **Understand that the 180° arc is half of the full circle's circumference.**
4. **Calculate the arc length:**
\[
\text{Arc Length} = \frac{180^\circ}{360^\circ} \times \text{Circumference}
\]
\[
\text{Arc Length} = \frac{1}{2} \times 6 \pi \text{ miles} = 3 \pi \text{ miles}
\]
**Answer:**
The length of a 180° arc is \( 3 \pi \) miles.
Please provide your final answer in the input box:
**Answer:**
\( \_\_\_\_\_\_\_\_ \text{ miles} \)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3e360c08-73c4-42e6-a160-35420a0e2834%2F138deef0-3074-4c94-908c-84aa8660c787%2F0g0alkm_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Understanding Arc Length**
**Problem Statement:**
The radius of a circle is 3 miles. What is the length of a 180° arc?
**Diagram Explanation:**
The provided diagram represents a circle with a radius (r) of 3 miles. The diagram highlights a 180° arc, which is half of the circle's circumference.
**Steps to Solve:**
1. **Identify the formula for the circumference of a circle:**
\[
\text{Circumference} = 2 \pi r
\]
2. **Substitute the given radius into the formula:**
\[
\text{Circumference} = 2 \pi \times 3 \text{ miles} = 6 \pi \text{ miles}
\]
3. **Understand that the 180° arc is half of the full circle's circumference.**
4. **Calculate the arc length:**
\[
\text{Arc Length} = \frac{180^\circ}{360^\circ} \times \text{Circumference}
\]
\[
\text{Arc Length} = \frac{1}{2} \times 6 \pi \text{ miles} = 3 \pi \text{ miles}
\]
**Answer:**
The length of a 180° arc is \( 3 \pi \) miles.
Please provide your final answer in the input box:
**Answer:**
\( \_\_\_\_\_\_\_\_ \text{ miles} \)
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