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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Transcribed Image Text:Below is a transcription of the image ideal for an educational website:
---
### Multiple-Choice Question on Angular Measurements and Distances
Please select the correct answer from the options provided:
- ○ 67.5 degrees
- ○ 8π km
- ○ 4π km
- ● 135 degrees
Note: This question includes both angular measurements (in degrees) and distances (in kilometers represented with the mathematical constant π).
---
The image showcases a multiple-choice question, listing four answers. The first answer is "67.5 degrees," the second is "8π kilometers," the third is "4π kilometers," and the fourth is "135 degrees." The correct answer, "135 degrees," is highlighted and marked with a filled-in circle.
![**Educational Website Content**
**Problem Statement:**
The diameter of a circle is 8 kilometers. What is the angle measure of an arc \(3\pi\) kilometers long?
**Diagram Explanation:**
There is a diagram of a circle. The circle is divided into two sectors, one shaded in blue and the other in yellow.
- The diameter (\(d\)) of the circle is labeled as 8 kilometers.
- The length of the arc (\(l\)) in the blue sector is labeled as \(3\pi\) kilometers.
**Question:**
Give the exact answer in simplest form.
**Solution Box:**
There is a box provided where the exact answer should be entered.
---
To solve this problem, we will use the relationship between the arc length and the angle subtended by the arc at the center of the circle. The formula for the arc length (\(l\)) is given by:
\[ l = r \theta \]
where \(r\) is the radius of the circle and \(\theta\) is the central angle in radians.
**Step-by-Step Solution:**
1. **Find the Radius:** The diameter (\(d\)) is 8 kilometers. The radius (\(r\)) is half the diameter:
\[ r = \frac{d}{2} = \frac{8}{2} = 4 \text{ kilometers} \]
2. **Set up the Relationship:** Using the arc length formula:
\[ l = r \theta \]
3. **Substitute the Given Values:**
The arc length (\(l\)) is \(3\pi\) kilometers, and the radius (\(r\)) is 4 kilometers. Hence:
\[ 3\pi = 4 \theta \]
4. **Solve for \(\theta\):**
\[ \theta = \frac{3\pi}{4} \]
Thus, the angle measure of the arc \(3\pi\) kilometers long is \(\frac{3\pi}{4}\) radians.
**Answer:**
\[
\boxed{\frac{3\pi}{4}}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F653d05cd-d0ff-49df-b258-3aae8e1a38c4%2F4a2f4607-544d-4dc2-9a5e-7ceddbbfd598%2F5s48p28_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Educational Website Content**
**Problem Statement:**
The diameter of a circle is 8 kilometers. What is the angle measure of an arc \(3\pi\) kilometers long?
**Diagram Explanation:**
There is a diagram of a circle. The circle is divided into two sectors, one shaded in blue and the other in yellow.
- The diameter (\(d\)) of the circle is labeled as 8 kilometers.
- The length of the arc (\(l\)) in the blue sector is labeled as \(3\pi\) kilometers.
**Question:**
Give the exact answer in simplest form.
**Solution Box:**
There is a box provided where the exact answer should be entered.
---
To solve this problem, we will use the relationship between the arc length and the angle subtended by the arc at the center of the circle. The formula for the arc length (\(l\)) is given by:
\[ l = r \theta \]
where \(r\) is the radius of the circle and \(\theta\) is the central angle in radians.
**Step-by-Step Solution:**
1. **Find the Radius:** The diameter (\(d\)) is 8 kilometers. The radius (\(r\)) is half the diameter:
\[ r = \frac{d}{2} = \frac{8}{2} = 4 \text{ kilometers} \]
2. **Set up the Relationship:** Using the arc length formula:
\[ l = r \theta \]
3. **Substitute the Given Values:**
The arc length (\(l\)) is \(3\pi\) kilometers, and the radius (\(r\)) is 4 kilometers. Hence:
\[ 3\pi = 4 \theta \]
4. **Solve for \(\theta\):**
\[ \theta = \frac{3\pi}{4} \]
Thus, the angle measure of the arc \(3\pi\) kilometers long is \(\frac{3\pi}{4}\) radians.
**Answer:**
\[
\boxed{\frac{3\pi}{4}}
\]
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