Mathematics For Machine Technology
8th Edition
ISBN:9781337798310
Author:Peterson, John.
Publisher:Peterson, John.
Chapter62: Volumes Of Prisms And Cylinders
Section: Chapter Questions
Problem 38A: A copper casting is in the shape of a prism with an equilateral triangle base. The length of each...
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Question
![### Geometry Problem
#### Question 10
**Problem Statement:**
What is the area of this shape?
(Shape is displayed with specific measurements.)
**Dimensions of the Shape:**
- The combined shape consists of a rectangle and a semicircle on top of it.
- Height of the rectangle: 3 cm
- Width of the rectangle: 2 cm
- Radius of the semicircle: 2 cm (as it shares the same width as the rectangle)
- Height of the full shape (inclusive of the semicircle): 6 cm
**Diagram Details:**
The diagram shows a vertical rectangle with a semicircle on top of it. The rectangle is 3 cm tall and 2 cm wide. The semicircle, sitting on top of the rectangle, has a radius of 2 cm.
**Answer Options:**
A. 83.52 cm²
B. 55.26 cm²
C. 64.26 cm²
D. 36 cm²
---
### Detailed Analysis:
To solve this problem, we need to calculate the area of the rectangle and add it to the area of the semicircle.
1. **Area of the Rectangle:**
\[ \text{Area of Rectangle} = \text{width} \times \text{height} \]
\[ \text{Area of Rectangle} = 2 \, \text{cm} \times 3 \, \text{cm} = 6 \, \text{cm}^2 \]
2. **Area of the Semicircle:**
- The formula for the area of a circle is \( \pi r^2 \), and since it’s a semicircle:
\[ \text{Area of Semicircle} = \frac{1}{2} \pi r^2 \]
\[ \text{Area of Semicircle} = \frac{1}{2} \pi (2 \, \text{cm})^2 = \frac{1}{2} \pi (4) \]
\[ \text{Area of Semicircle} = 2 \pi \, \text{cm}^2 \approx 6.28 \, \text{cm}^2 \]
3. **Total Area of the Shape:**
\[ \text{Total Area} = \text{Area of Rectangle} + \text{](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3b4ac02d-8fa9-4cd1-ad8a-49ffeef4197d%2Ffe9ec05d-7970-4f06-b25f-8a9efd33cac9%2Ftr6jm2_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Geometry Problem
#### Question 10
**Problem Statement:**
What is the area of this shape?
(Shape is displayed with specific measurements.)
**Dimensions of the Shape:**
- The combined shape consists of a rectangle and a semicircle on top of it.
- Height of the rectangle: 3 cm
- Width of the rectangle: 2 cm
- Radius of the semicircle: 2 cm (as it shares the same width as the rectangle)
- Height of the full shape (inclusive of the semicircle): 6 cm
**Diagram Details:**
The diagram shows a vertical rectangle with a semicircle on top of it. The rectangle is 3 cm tall and 2 cm wide. The semicircle, sitting on top of the rectangle, has a radius of 2 cm.
**Answer Options:**
A. 83.52 cm²
B. 55.26 cm²
C. 64.26 cm²
D. 36 cm²
---
### Detailed Analysis:
To solve this problem, we need to calculate the area of the rectangle and add it to the area of the semicircle.
1. **Area of the Rectangle:**
\[ \text{Area of Rectangle} = \text{width} \times \text{height} \]
\[ \text{Area of Rectangle} = 2 \, \text{cm} \times 3 \, \text{cm} = 6 \, \text{cm}^2 \]
2. **Area of the Semicircle:**
- The formula for the area of a circle is \( \pi r^2 \), and since it’s a semicircle:
\[ \text{Area of Semicircle} = \frac{1}{2} \pi r^2 \]
\[ \text{Area of Semicircle} = \frac{1}{2} \pi (2 \, \text{cm})^2 = \frac{1}{2} \pi (4) \]
\[ \text{Area of Semicircle} = 2 \pi \, \text{cm}^2 \approx 6.28 \, \text{cm}^2 \]
3. **Total Area of the Shape:**
\[ \text{Total Area} = \text{Area of Rectangle} + \text{
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