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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Transcribed Image Text:## #22. Find the Volume of the Solid Below
You are provided with an image of a composite solid which appears to be made up of rectangular prisms arranged in a "T" shape. The dimensions of the solid are labeled as follows:
- The length of the top horizontal part of the "T" is 18 cm.
- The height of the solid (equal on both sides of the vertical stem) is 3 cm.
- The width of the horizontal part of the "T" is 3 cm.
- The stem of the "T" directly below the horizontal top has a horizontal length of 6 cm.
- The width of the vertical part of the "T" is labeled as 1.5 cm.
To find the volume of the solid, it can be broken down into smaller, easily measurable prisms, and then the volumes of these prisms can be added together.
![# Volume Calculation of a 3D Solid
## Problem Statement
**Question 22**: Find the volume of the solid below.
## Description of the Diagram:
The diagram illustrates a 3-dimensional solid object that is a composite of two rectangular prisms:
- Overall dimensions: The solid has a length of 18 cm, a width of 6 cm, and heights of 3 cm in two sections.
- Segment details: There are two intersecting rectangular prisms.
- The first rectangular prism, oriented horizontally, has dimensions of 18 cm (length) x 3 cm (width) x 3 cm (height).
- The second rectangular prism is oriented perpendicular to the first and intersects it. It has dimensions of 15 cm (length) x 3 cm (width) x 3 cm (height).
- The intersection depth, where one prism overlaps into the other, affects the calculation of the total volume.
## Volume Calculation:
To find the volume of the entire solid, we will calculate the volume of each rectangular prism and subtract the overlapping volume.
1. **Volume of the first prism**:
\[
V_1 = \text{length} \times \text{width} \times \text{height} = 18 \, \text{cm} \times 3 \, \text{cm} \times 3 \, \text{cm} = 162 \, \text{cm}^3
\]
2. **Volume of the second prism**:
\[
V_2 = \text{length} \times \text{width} \times \text{height} = 15 \, \text{cm} \times 3 \, \text{cm} \times 3 \, \text{cm} = 135 \, \text{cm}^3
\]
3. **Overlapping volume**:
The overlapping volume should be one of the portions that both prisms share. The intersection volume is calculated by:
\[
V_{\text{overlap}} = \text{length} \times \text{width} \times \text{height} = 3 \, \text{cm} \times 3 \, \text{cm} \times 3 \, \text{cm} = 27 \, \text{cm}^3
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9bd23e95-b462-4379-b394-ae7cf6f33191%2F55d9d746-ea89-4888-8510-942ae6d03c98%2Fieh0owb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:# Volume Calculation of a 3D Solid
## Problem Statement
**Question 22**: Find the volume of the solid below.
## Description of the Diagram:
The diagram illustrates a 3-dimensional solid object that is a composite of two rectangular prisms:
- Overall dimensions: The solid has a length of 18 cm, a width of 6 cm, and heights of 3 cm in two sections.
- Segment details: There are two intersecting rectangular prisms.
- The first rectangular prism, oriented horizontally, has dimensions of 18 cm (length) x 3 cm (width) x 3 cm (height).
- The second rectangular prism is oriented perpendicular to the first and intersects it. It has dimensions of 15 cm (length) x 3 cm (width) x 3 cm (height).
- The intersection depth, where one prism overlaps into the other, affects the calculation of the total volume.
## Volume Calculation:
To find the volume of the entire solid, we will calculate the volume of each rectangular prism and subtract the overlapping volume.
1. **Volume of the first prism**:
\[
V_1 = \text{length} \times \text{width} \times \text{height} = 18 \, \text{cm} \times 3 \, \text{cm} \times 3 \, \text{cm} = 162 \, \text{cm}^3
\]
2. **Volume of the second prism**:
\[
V_2 = \text{length} \times \text{width} \times \text{height} = 15 \, \text{cm} \times 3 \, \text{cm} \times 3 \, \text{cm} = 135 \, \text{cm}^3
\]
3. **Overlapping volume**:
The overlapping volume should be one of the portions that both prisms share. The intersection volume is calculated by:
\[
V_{\text{overlap}} = \text{length} \times \text{width} \times \text{height} = 3 \, \text{cm} \times 3 \, \text{cm} \times 3 \, \text{cm} = 27 \, \text{cm}^3
\]
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