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
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![**Problem Statement:**
Find the length of the arc. Round your answer to the nearest tenth.
**Diagram Explanation:**
The diagram shows a partial circle (arc) with a radius of 12 meters. The central angle that subtends the arc is 270°.
**Solution:**
To find the length of the arc (L), we use the formula for the arc length of a circle, which is:
\[ L = 2 \pi r \left( \frac{\theta}{360} \right) \]
where:
- \( r \) is the radius of the circle.
- \( \theta \) is the central angle in degrees.
In this problem:
- \( r = 12 \) meters
- \( \theta = 270° \)
Substitute the values into the formula:
\[ L = 2 \pi \times 12 \left( \frac{270}{360} \right) \]
First, simplify the fraction:
\[ \frac{270}{360} = \frac{3}{4} \]
Now substitute back into the formula:
\[ L = 2 \pi \times 12 \times \frac{3}{4} = 24 \pi \times \frac{3}{4} = 18 \pi \]
Next, multiply by \( \pi \approx 3.14 \):
\[ L \approx 18 \times 3.14 = 56.52 \]
Round to the nearest tenth:
\[ \boxed{56.5 \text{ meters}} \]
So, the length of the arc is approximately 56.5 meters.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F38119a94-a193-4725-8b44-bc4499933554%2Ff75c446f-c785-45f1-ba6a-81523f9e177e%2Fmw76gl_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the length of the arc. Round your answer to the nearest tenth.
**Diagram Explanation:**
The diagram shows a partial circle (arc) with a radius of 12 meters. The central angle that subtends the arc is 270°.
**Solution:**
To find the length of the arc (L), we use the formula for the arc length of a circle, which is:
\[ L = 2 \pi r \left( \frac{\theta}{360} \right) \]
where:
- \( r \) is the radius of the circle.
- \( \theta \) is the central angle in degrees.
In this problem:
- \( r = 12 \) meters
- \( \theta = 270° \)
Substitute the values into the formula:
\[ L = 2 \pi \times 12 \left( \frac{270}{360} \right) \]
First, simplify the fraction:
\[ \frac{270}{360} = \frac{3}{4} \]
Now substitute back into the formula:
\[ L = 2 \pi \times 12 \times \frac{3}{4} = 24 \pi \times \frac{3}{4} = 18 \pi \]
Next, multiply by \( \pi \approx 3.14 \):
\[ L \approx 18 \times 3.14 = 56.52 \]
Round to the nearest tenth:
\[ \boxed{56.5 \text{ meters}} \]
So, the length of the arc is approximately 56.5 meters.
![**Finding the Length of an Arc**
In this example, we are given a circle with a radius of 4 cm and a central angle of 60°. We need to find the length of the arc corresponding to this angle, and then round the answer to the nearest tenth.
### Step-by-Step Explanation
1. **Identify the given information:**
- Radius (r) = 4 cm
- Central angle (θ) = 60°
2. **Formula for the Arc Length:**
The formula to calculate the length of an arc (L) is:
\[
L = \frac{\theta}{360°} \times 2\pi r
\]
where θ is the central angle in degrees and r is the radius of the circle.
3. **Plug in the values:**
- θ = 60°
- r = 4 cm
Substitute these values into the formula:
\[
L = \frac{60}{360} \times 2\pi \times 4
\]
4. **Simplify the expression:**
\[
L = \frac{1}{6} \times 8\pi
\]
5. **Calculate the arc length:**
\[
L = \frac{8\pi}{6}
\]
\[
L = \frac{4\pi}{3}
\]
6. **Approximate using the value of π (pi ≈ 3.14):**
\[
L = \frac{4 \times 3.14}{3}
\]
\[
L \approx \frac{12.56}{3}
\]
\[
L \approx 4.2 \, \text{cm}
\]
### Conclusion
The length of the arc, rounded to the nearest tenth, is approximately **4.2 cm**.
### Explanation of Diagram
The diagram shows a circle with a radius of 4 cm. A sector of the circle is marked, defined by a central angle of 60°. The arc length of this sector is the part of the circumference that we are calculating.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F38119a94-a193-4725-8b44-bc4499933554%2Ff75c446f-c785-45f1-ba6a-81523f9e177e%2Fm399l9k_processed.png&w=3840&q=75)
Transcribed Image Text:**Finding the Length of an Arc**
In this example, we are given a circle with a radius of 4 cm and a central angle of 60°. We need to find the length of the arc corresponding to this angle, and then round the answer to the nearest tenth.
### Step-by-Step Explanation
1. **Identify the given information:**
- Radius (r) = 4 cm
- Central angle (θ) = 60°
2. **Formula for the Arc Length:**
The formula to calculate the length of an arc (L) is:
\[
L = \frac{\theta}{360°} \times 2\pi r
\]
where θ is the central angle in degrees and r is the radius of the circle.
3. **Plug in the values:**
- θ = 60°
- r = 4 cm
Substitute these values into the formula:
\[
L = \frac{60}{360} \times 2\pi \times 4
\]
4. **Simplify the expression:**
\[
L = \frac{1}{6} \times 8\pi
\]
5. **Calculate the arc length:**
\[
L = \frac{8\pi}{6}
\]
\[
L = \frac{4\pi}{3}
\]
6. **Approximate using the value of π (pi ≈ 3.14):**
\[
L = \frac{4 \times 3.14}{3}
\]
\[
L \approx \frac{12.56}{3}
\]
\[
L \approx 4.2 \, \text{cm}
\]
### Conclusion
The length of the arc, rounded to the nearest tenth, is approximately **4.2 cm**.
### Explanation of Diagram
The diagram shows a circle with a radius of 4 cm. A sector of the circle is marked, defined by a central angle of 60°. The arc length of this sector is the part of the circumference that we are calculating.
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