Select all shapes that have a volume of 36 π.

Mathematics For Machine Technology
8th Edition
ISBN:9781337798310
Author:Peterson, John.
Publisher:Peterson, John.
Chapter63: Volumes Of Pyramids And Cones
Section: Chapter Questions
Problem 14A
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Select all shapes that have a volume of

36 π. 

**Understanding the Dimensions of a Cone**

In the image, we see a right circular cone. The essential characteristics of this cone are clearly depicted with the following measurements:

- **Height (h):** The perpendicular distance from the base to the apex (top) of the cone is \(12\) units.
- **Radius (r):** The radius of the base of the cone, which is the distance from the center of the base to any point on its circumference, is \(3\) units.

### Diagram Explanation:

The cone is illustrated in a two-dimensional diagram. A dashed line represents the height of the cone, running vertically from the base to the tip. The radius is marked as extending horizontally from the center to the edge of the base, forming a right angle with the height.

**Option:**
- Option 5

**Next Steps:**

To further explore this cone, you could calculate various properties such as the volume and surface area. Use the following formulas:

- **Volume (V) of a cone:**
  \[ V = \frac{1}{3} \pi r^2 h \]
- **Surface Area (A) of a cone:**
  \[ A = \pi r (r + \sqrt{h^2 + r^2}) \]

Where \( \pi \) (Pi) is approximately \( 3.14159 \).

In our example:
- \( r = 3 \)
- \( h = 12 \)

By substituting these values into the formulas, you can find the specific volume and surface area of the cone depicted in the diagram.
Transcribed Image Text:**Understanding the Dimensions of a Cone** In the image, we see a right circular cone. The essential characteristics of this cone are clearly depicted with the following measurements: - **Height (h):** The perpendicular distance from the base to the apex (top) of the cone is \(12\) units. - **Radius (r):** The radius of the base of the cone, which is the distance from the center of the base to any point on its circumference, is \(3\) units. ### Diagram Explanation: The cone is illustrated in a two-dimensional diagram. A dashed line represents the height of the cone, running vertically from the base to the tip. The radius is marked as extending horizontally from the center to the edge of the base, forming a right angle with the height. **Option:** - Option 5 **Next Steps:** To further explore this cone, you could calculate various properties such as the volume and surface area. Use the following formulas: - **Volume (V) of a cone:** \[ V = \frac{1}{3} \pi r^2 h \] - **Surface Area (A) of a cone:** \[ A = \pi r (r + \sqrt{h^2 + r^2}) \] Where \( \pi \) (Pi) is approximately \( 3.14159 \). In our example: - \( r = 3 \) - \( h = 12 \) By substituting these values into the formulas, you can find the specific volume and surface area of the cone depicted in the diagram.
### Volume Selection Exercise

**Instructions:**
Select all shapes that have a volume of 36π.

#### Options:

**Option 1: Cone**
- Diagram: A cone with a height of 6 units and a radius of 6 units.
- Volume Formula: \(\frac{1}{3}\pi r^2 h\)

**Option 2: Triangular Prism**
- Diagram: A prism with a triangular base, height of 3 units, base length of 4 units, and a width of \(3\pi\) units.
- Volume Formula: \(\frac{1}{2} \times \text{Base length} \times \text{Height} \times \text{Prism length}\)

**Option 3: Rectangular Prism**
- Diagram: A rectangular prism with dimensions 3 units \(\times\) \(3\pi\) units \(\times\) 4 units.
- Volume Formula: \( \text{Length} \times \text{Width} \times \text{Height}\)

**Option 4: Pyramid**
- Diagram: A pyramid with a square base of side length 4 units and height of 3 units. The pyramid height is 3 units.
- Volume Formula: \(\frac{1}{3} \times \text{Base Area} \times \text{Height}\)

**Calculations and Explanations:**
1. **Option 1: Cone**
   - Calculate the volume: \(\frac{1}{3}\pi \times 6^2 \times 6 = 72\pi\)
   - The volume is 72π, not 36π.

2. **Option 2: Triangular Prism**
   - Calculate the volume: \(\frac{1}{2} \times 4 \times 3 \times 3\pi = 18\pi\)
   - The volume is 18π, not 36π.

3. **Option 3: Rectangular Prism**
   - Calculate the volume: \(3 \times 4 \times 3\pi = 36\pi\)
   - The volume is 36π.

4. **Option 4: Pyramid**
   - Calculate the volume: \(\frac{1}{3} \times (4^2) \times 3 = 16\)
   - The volume is 16, not including
Transcribed Image Text:### Volume Selection Exercise **Instructions:** Select all shapes that have a volume of 36π. #### Options: **Option 1: Cone** - Diagram: A cone with a height of 6 units and a radius of 6 units. - Volume Formula: \(\frac{1}{3}\pi r^2 h\) **Option 2: Triangular Prism** - Diagram: A prism with a triangular base, height of 3 units, base length of 4 units, and a width of \(3\pi\) units. - Volume Formula: \(\frac{1}{2} \times \text{Base length} \times \text{Height} \times \text{Prism length}\) **Option 3: Rectangular Prism** - Diagram: A rectangular prism with dimensions 3 units \(\times\) \(3\pi\) units \(\times\) 4 units. - Volume Formula: \( \text{Length} \times \text{Width} \times \text{Height}\) **Option 4: Pyramid** - Diagram: A pyramid with a square base of side length 4 units and height of 3 units. The pyramid height is 3 units. - Volume Formula: \(\frac{1}{3} \times \text{Base Area} \times \text{Height}\) **Calculations and Explanations:** 1. **Option 1: Cone** - Calculate the volume: \(\frac{1}{3}\pi \times 6^2 \times 6 = 72\pi\) - The volume is 72π, not 36π. 2. **Option 2: Triangular Prism** - Calculate the volume: \(\frac{1}{2} \times 4 \times 3 \times 3\pi = 18\pi\) - The volume is 18π, not 36π. 3. **Option 3: Rectangular Prism** - Calculate the volume: \(3 \times 4 \times 3\pi = 36\pi\) - The volume is 36π. 4. **Option 4: Pyramid** - Calculate the volume: \(\frac{1}{3} \times (4^2) \times 3 = 16\) - The volume is 16, not including
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