Trigonometry (11th Edition)
11th Edition
ISBN: 9780134217437
Author: Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher: PEARSON
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### Analysis and Solution
To solve for the shaded area within the rectangle AFEB that includes a semicircle, we need to follow these steps:
1. **Calculate the area of the rectangle AFEB:**
Given the dimensions:
- Height of the rectangle (AB): 5 inches
- Width of the rectangle (CD = FE): 6 inches
Area of the rectangle \(A_{rectangle}\):
\[ A_{rectangle} = \text{Height} \times \text{Width} = 5 \times 6 = 30 \, \text{square inches} \]
2. **Calculate the area of the semicircle with diameter CD:**
To find the area of the semicircle, we first find the area of a full circle with diameter CD:
- Diameter (CD): 6 inches
- Radius (r): \( \frac{\text{Diameter}}{2} = \frac{6}{2} = 3 \, \text{inches} \)
Area of the full circle:
\[ A_{circle} = \pi r^2 = \pi (3^2) = 9\pi \, \text{square inches} \]
Area of the semicircle \(A_{semicircle}\):
\[ A_{semicircle} = \frac{A_{circle}}{2} = \frac{9\pi}{2} \approx \frac{28.27}{2} = 14.14 \, \text{square inches} \]
3. **Calculate the area of the shaded region (rectangle minus semicircle):**
\[ A_{shaded} = A_{rectangle} - A_{semicircle} = 30 - 14.14 = 15.86 \, \text{square inches} \]
### Answer:
The area of the shaded region is approximately **15.86 square inches**.](https://content.bartleby.com/qna-images/question/87804e42-439c-4b15-9fec-573b7a02348d/ab26ff93-b67a-4bbe-8b59-592e78b67feb/l93igsf_processed.jpeg)
Transcribed Image Text:### Problem Description
In the diagram below of rectangle AFEB and a semicircle with diameter CD, AB = 5 inches, AB = BC = DE = FE, and CD = 6 inches. Find the area of the shaded region, to the nearest hundredth of a square inch.
### Analysis and Solution
To solve for the shaded area within the rectangle AFEB that includes a semicircle, we need to follow these steps:
1. **Calculate the area of the rectangle AFEB:**
Given the dimensions:
- Height of the rectangle (AB): 5 inches
- Width of the rectangle (CD = FE): 6 inches
Area of the rectangle \(A_{rectangle}\):
\[ A_{rectangle} = \text{Height} \times \text{Width} = 5 \times 6 = 30 \, \text{square inches} \]
2. **Calculate the area of the semicircle with diameter CD:**
To find the area of the semicircle, we first find the area of a full circle with diameter CD:
- Diameter (CD): 6 inches
- Radius (r): \( \frac{\text{Diameter}}{2} = \frac{6}{2} = 3 \, \text{inches} \)
Area of the full circle:
\[ A_{circle} = \pi r^2 = \pi (3^2) = 9\pi \, \text{square inches} \]
Area of the semicircle \(A_{semicircle}\):
\[ A_{semicircle} = \frac{A_{circle}}{2} = \frac{9\pi}{2} \approx \frac{28.27}{2} = 14.14 \, \text{square inches} \]
3. **Calculate the area of the shaded region (rectangle minus semicircle):**
\[ A_{shaded} = A_{rectangle} - A_{semicircle} = 30 - 14.14 = 15.86 \, \text{square inches} \]
### Answer:
The area of the shaded region is approximately **15.86 square inches**.
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