A square based prism and a cylinder both have the same height of 4 cm and the same base area. If the volume of the square based prism is 452 cm³, based on the concepts of Cavalieri's principle, what is the approximate radius of the base of the cylinder? Volume Prism = Bh B Circle=1₁² 4 cm A 21 cm B 43 cm. 6 cm U
A square based prism and a cylinder both have the same height of 4 cm and the same base area. If the volume of the square based prism is 452 cm³, based on the concepts of Cavalieri's principle, what is the approximate radius of the base of the cylinder? Volume Prism = Bh B Circle=1₁² 4 cm A 21 cm B 43 cm. 6 cm U
Mathematics For Machine Technology
8th Edition
ISBN:9781337798310
Author:Peterson, John.
Publisher:Peterson, John.
Chapter65: Achievement Review—section Six
Section: Chapter Questions
Problem 55AR: Solve these prism and cylinder exercises. Where necessary, round the answers to 2 decimal places...
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![### Understanding Cavalieri’s Principle to Find the Radius of a Cylinder
A square-based prism and a cylinder both have the same height of 4 cm and the same base area. If the volume of the square-based prism is 452 cm³, based on the concepts of Cavalieri's principle, what is the approximate radius of the base of the cylinder?
**Formulas:**
- Volume of Prism \( V_{\text{Prism}} = Bh \)
- Area of Circle \( B_{\text{Circle}} = \pi r^2 \)
#### Problem Illustration
The diagram presents a square-based prism and a cylinder with the same height (4 cm) placed side by side to visually suggest they have the same volume.
- **Cylinder**: Illustrated as a 3D shape with a circular base.
- **Square-based Prism**: Illustrated as a 3D shape with a square base.
- Both shapes have a height of 4 cm.
#### Given Data
- Height (\( h \)) of both shapes = 4 cm
- Volume (\( V_{\text{Prism}} \)) of the prism = 452 cm³
#### Goal
To determine the radius (\( r \)) of the base of the cylinder.
#### Step-by-Step Solution:
1. **Volume of the Prism:**
\[
V_{\text{Prism}} = Bh
\]
Where \( B \) = base area and \( h \) = height.
2. **Calculate the Base Area of the Prism:**
Given:
\[
V_{\text{Prism}} = 452 \text{ cm}^3
\]
\[
h = 4 \text{ cm}
\]
Solving for \( B \):
\[
452 = B \times 4
\]
\[
B = \frac{452}{4} = 113 \text{ cm}^2
\]
3. **Use the Base Area for the Cylinder:**
The base area of the cylinder is the same as the base area of the prism.
\[
B_{\text{Circle}} = \pi r^2 = 113 \text{ cm}^2
\]
4. **Solve for the Radius \( r \):**
\[
r^2 = \frac{113}{\pi}
\](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F95b555fe-c61f-4380-ac77-d5f60f2cc03d%2F864f6145-fdf7-4b29-b498-01bda7ca0cc3%2Fg9d410w_processed.png&w=3840&q=75)
Transcribed Image Text:### Understanding Cavalieri’s Principle to Find the Radius of a Cylinder
A square-based prism and a cylinder both have the same height of 4 cm and the same base area. If the volume of the square-based prism is 452 cm³, based on the concepts of Cavalieri's principle, what is the approximate radius of the base of the cylinder?
**Formulas:**
- Volume of Prism \( V_{\text{Prism}} = Bh \)
- Area of Circle \( B_{\text{Circle}} = \pi r^2 \)
#### Problem Illustration
The diagram presents a square-based prism and a cylinder with the same height (4 cm) placed side by side to visually suggest they have the same volume.
- **Cylinder**: Illustrated as a 3D shape with a circular base.
- **Square-based Prism**: Illustrated as a 3D shape with a square base.
- Both shapes have a height of 4 cm.
#### Given Data
- Height (\( h \)) of both shapes = 4 cm
- Volume (\( V_{\text{Prism}} \)) of the prism = 452 cm³
#### Goal
To determine the radius (\( r \)) of the base of the cylinder.
#### Step-by-Step Solution:
1. **Volume of the Prism:**
\[
V_{\text{Prism}} = Bh
\]
Where \( B \) = base area and \( h \) = height.
2. **Calculate the Base Area of the Prism:**
Given:
\[
V_{\text{Prism}} = 452 \text{ cm}^3
\]
\[
h = 4 \text{ cm}
\]
Solving for \( B \):
\[
452 = B \times 4
\]
\[
B = \frac{452}{4} = 113 \text{ cm}^2
\]
3. **Use the Base Area for the Cylinder:**
The base area of the cylinder is the same as the base area of the prism.
\[
B_{\text{Circle}} = \pi r^2 = 113 \text{ cm}^2
\]
4. **Solve for the Radius \( r \):**
\[
r^2 = \frac{113}{\pi}
\
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