Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN: 9780134463216
Author: Robert F. Blitzer
Publisher: PEARSON
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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
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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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