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Let's model R2-D2's body with a cylinder of height 3 feet and radius 0.75 feet and head with a semi-sphere of radius 0.75 feet. What is the total volume of R2-D2's body and head? *

Let's model R2-D2's body with a cylinder of height 3 feet and radius 0.75 feet and head with a semi-sphere of radius 0.75 feet. What is the total volume of R2-D2's body and head? *

Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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Let’s model R2-D2’s body with a cylinder of height 3 feet and radius 0.75 feet and head with a semi-sphere of radius 0.75 feet. What is the total volume of R2-D2’s body and head? 

 

**Modeling R2-D2's Body and Head: Calculating Volume** Let's model R2-D2's body with a cylinder of height 3 feet and radius 0.75 feet and head with a semi-sphere of radius 0.75 feet. What is the total volume of R2-D2's body and head? **Diagram Explanation:** - The diagram shows a cylinder with a semi-sphere on top. - The cylinder has a radius (r) of 0.75 feet and a height (h) of 3 feet. - The semi-sphere, which represents R2-D2's head, also has a radius (r) of 0.75 feet. ### Calculations: 1. **Volume of the Cylinder:** The formula for the volume of a cylinder is \(V = \pi r^2 h\). Substituting the given values: \[ V_{\text{cylinder}} = \pi \times (0.75)^2 \times 3 \] 2. **Volume of the Semi-Sphere:** The formula for the volume of a sphere is \(V = \frac{4}{3} \pi r^3\), and for a semi-sphere it is \(\frac{1}{2}\) of that. Substituting the given values: \[ V_{\text{semi-sphere}} = \frac{1}{2} \times \frac{4}{3} \pi \times (0.75)^3 \] 3. **Total Volume:** Sum the volumes of the cylinder and the semi-sphere. \[ V_{\text{total}} = V_{\text{cylinder}} + V_{\text{semi-sphere}} \] By calculating the above formulas, we get the total volume of R2-D2’s body and head. These calculations allow us to model R2-D2’s volume as a combination of simple geometric shapes, which can be useful in various educational and engineering contexts.
Transcribed Image Text:**Modeling R2-D2's Body and Head: Calculating Volume** Let's model R2-D2's body with a cylinder of height 3 feet and radius 0.75 feet and head with a semi-sphere of radius 0.75 feet. What is the total volume of R2-D2's body and head? **Diagram Explanation:** - The diagram shows a cylinder with a semi-sphere on top. - The cylinder has a radius (r) of 0.75 feet and a height (h) of 3 feet. - The semi-sphere, which represents R2-D2's head, also has a radius (r) of 0.75 feet. ### Calculations: 1. **Volume of the Cylinder:** The formula for the volume of a cylinder is \(V = \pi r^2 h\). Substituting the given values: \[ V_{\text{cylinder}} = \pi \times (0.75)^2 \times 3 \] 2. **Volume of the Semi-Sphere:** The formula for the volume of a sphere is \(V = \frac{4}{3} \pi r^3\), and for a semi-sphere it is \(\frac{1}{2}\) of that. Substituting the given values: \[ V_{\text{semi-sphere}} = \frac{1}{2} \times \frac{4}{3} \pi \times (0.75)^3 \] 3. **Total Volume:** Sum the volumes of the cylinder and the semi-sphere. \[ V_{\text{total}} = V_{\text{cylinder}} + V_{\text{semi-sphere}} \] By calculating the above formulas, we get the total volume of R2-D2’s body and head. These calculations allow us to model R2-D2’s volume as a combination of simple geometric shapes, which can be useful in various educational and engineering contexts.
For a sphere with radius, \( r \), its volume is \( \frac{4}{3} \pi r^3 \).
Transcribed Image Text:For a sphere with radius, \( r \), its volume is \( \frac{4}{3} \pi r^3 \).
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