Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)
6th Edition
ISBN:9781337111348
Author:Bruce Crauder, Benny Evans, Alan Noell
Publisher:Bruce Crauder, Benny Evans, Alan Noell
Chapter2: Graphical And Tabular Analysis
Section2.6: Optimization
Problem 16E: Laying Phone Cable City A lies on the north bank of a river that is 1 mile wide. You need to run a...
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![**Problem 5: Geometry - Area of Shaded Region**
**Description:**
The given image is a geometry problem. Below it is a circle with a radius of 4.2 meters. A sector of the circle is shaded. The central angle of the sector is not explicitly given, but it appears to be a portion of the circle.
**Step-by-Step Solution:**
1. **Identify the Radius:**
- The radius of the circle is \( 4.2 \) meters.
2. **Calculate the Full Circle Area:**
- The area \( A \) of a circle is given by the formula:
\[
A = \pi r^2
\]
- Substitute \( r = 4.2 \):
\[
A = \pi (4.2)^2 = \pi \times 17.64 \approx 55.42 \, \text{square meters}
\]
3. **Determine the Fraction of the Circle:**
- Visually estimate the fraction of the shaded sector. If the angle is not given, one typically assumes some common angles, like \( 90^\circ \) (quarter circle), \( 180^\circ \) (half circle), etc. Here, it appears the shaded region is roughly a quarter of the circle.
4. **Calculate the Area of the Shaded Region:**
- If the shaded region is indeed a quarter circle, the area of the shaded region \( A_s \) is:
\[
A_s = \frac{1}{4} \times 55.42 \approx 13.86 \, \text{square meters}
\]
**Graph/Diagram Explanation:**
- The circle shown has a single sector shaded. The sector starts from the center and extends to a part of the circumference creating an angle at the center.
- The radius of the circle (4.2 meters) is indicated within the circle, specifically pointing to the edge of the sector.
- Assuming the shaded region is a quarter of the circle (based on visual estimation):
Thus, the area of the shaded region is approximately 13.86 square meters.
**Disclaimer:**
The accuracy of the shaded area calculation depends on the precise central angle of the sector, which might not be explicitly clear from the image. Proper determination requires knowing or measuring the central angle.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8c8a549a-ce13-4c73-8e46-ff62e8d935c2%2Faa60a87b-d4b4-4c8a-9e38-68051aa6abbe%2Frhedr8b_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 5: Geometry - Area of Shaded Region**
**Description:**
The given image is a geometry problem. Below it is a circle with a radius of 4.2 meters. A sector of the circle is shaded. The central angle of the sector is not explicitly given, but it appears to be a portion of the circle.
**Step-by-Step Solution:**
1. **Identify the Radius:**
- The radius of the circle is \( 4.2 \) meters.
2. **Calculate the Full Circle Area:**
- The area \( A \) of a circle is given by the formula:
\[
A = \pi r^2
\]
- Substitute \( r = 4.2 \):
\[
A = \pi (4.2)^2 = \pi \times 17.64 \approx 55.42 \, \text{square meters}
\]
3. **Determine the Fraction of the Circle:**
- Visually estimate the fraction of the shaded sector. If the angle is not given, one typically assumes some common angles, like \( 90^\circ \) (quarter circle), \( 180^\circ \) (half circle), etc. Here, it appears the shaded region is roughly a quarter of the circle.
4. **Calculate the Area of the Shaded Region:**
- If the shaded region is indeed a quarter circle, the area of the shaded region \( A_s \) is:
\[
A_s = \frac{1}{4} \times 55.42 \approx 13.86 \, \text{square meters}
\]
**Graph/Diagram Explanation:**
- The circle shown has a single sector shaded. The sector starts from the center and extends to a part of the circumference creating an angle at the center.
- The radius of the circle (4.2 meters) is indicated within the circle, specifically pointing to the edge of the sector.
- Assuming the shaded region is a quarter of the circle (based on visual estimation):
Thus, the area of the shaded region is approximately 13.86 square meters.
**Disclaimer:**
The accuracy of the shaded area calculation depends on the precise central angle of the sector, which might not be explicitly clear from the image. Proper determination requires knowing or measuring the central angle.
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