Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Question
Calculate The volume of the part to the nearest cubic CM.
![### Problem 10: Calculating Volume
#### Objective:
Calculate the volume of the part to the nearest cubic centimeter.
#### Diagram Description:
- The given illustration depicts a geometrical part with a rectangular base and a semi-circular top extension.
- The dimensions of the rectangular part are labeled as 10 cm (width) and 14 cm (height).
- The semi-circular top extension has a radius (R) of 2 cm.
- A central cylindrical hole extends through the entire height of the part, with a radius of 3 cm.
#### Steps to Calculate Volume:
To solve this problem, you need to calculate the volume of the composite shape created by combining the volume of the rectangular part and the semi-circular top extension, and then subtracting the volume of the cylindrical hole.
1. **Volume of the Rectangular Part (V_rect):**
\[ V_{rect} = \text{Length} \times \text{Width} \times \text{Height} \]
Here:
- Length = 14 cm
- Width = 10 cm
- Height = (assuming the part's thickness is not provided, we consider it extends to 1 cm)
\[ V_{rect} = 14 \times 10 \times thickness \]
2. **Volume of the Semi-Circular Extension (V_semi):**
\[ V_{semi} = \frac{1}{2} \times \pi \times R^2 \times Width \]
Here:
- Radius (R) = 2 cm
- Width = 10 cm
\[ V_{semi} = \frac{1}{2} \times \pi \times 2^2 \times 10 \]
3. **Volume of the Cylindrical Hole (V_hole):**
\[ V_{hole} = \pi \times r^2 \times Height \]
Here:
- Radius (r) = 3 cm
- Height = 14 cm
\[ V_{hole} = \pi \times 3^2 \times 14 \]
4. **Total Volume (V_total):**
\[ V_{total} = V_{rect} + V_{semi} - V_{hole} \]
Please proceed with the calculations substituting the given values and using the approximation \(\pi \approx 3.14](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0e9c998a-5af1-4bc2-b298-e1462b059995%2Faca73922-0d9d-4449-8349-dd654408a69c%2Fl7xpuk4_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem 10: Calculating Volume
#### Objective:
Calculate the volume of the part to the nearest cubic centimeter.
#### Diagram Description:
- The given illustration depicts a geometrical part with a rectangular base and a semi-circular top extension.
- The dimensions of the rectangular part are labeled as 10 cm (width) and 14 cm (height).
- The semi-circular top extension has a radius (R) of 2 cm.
- A central cylindrical hole extends through the entire height of the part, with a radius of 3 cm.
#### Steps to Calculate Volume:
To solve this problem, you need to calculate the volume of the composite shape created by combining the volume of the rectangular part and the semi-circular top extension, and then subtracting the volume of the cylindrical hole.
1. **Volume of the Rectangular Part (V_rect):**
\[ V_{rect} = \text{Length} \times \text{Width} \times \text{Height} \]
Here:
- Length = 14 cm
- Width = 10 cm
- Height = (assuming the part's thickness is not provided, we consider it extends to 1 cm)
\[ V_{rect} = 14 \times 10 \times thickness \]
2. **Volume of the Semi-Circular Extension (V_semi):**
\[ V_{semi} = \frac{1}{2} \times \pi \times R^2 \times Width \]
Here:
- Radius (R) = 2 cm
- Width = 10 cm
\[ V_{semi} = \frac{1}{2} \times \pi \times 2^2 \times 10 \]
3. **Volume of the Cylindrical Hole (V_hole):**
\[ V_{hole} = \pi \times r^2 \times Height \]
Here:
- Radius (r) = 3 cm
- Height = 14 cm
\[ V_{hole} = \pi \times 3^2 \times 14 \]
4. **Total Volume (V_total):**
\[ V_{total} = V_{rect} + V_{semi} - V_{hole} \]
Please proceed with the calculations substituting the given values and using the approximation \(\pi \approx 3.14
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