Question
Math Practice U7 L8
Unit 7 Math Practice Links
Live Lesson Recording
Function of the Day:
How would we find the volume of the pencil? What else would we need to know?
What do you know?
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Vocabulary Terms - Define/Illustrate
Describe the differences between a Sphere, a Cone, and a Cylinder?
How are the formulas for these circular solids related to each other?
5 Problems - SHOW ALL WORK!
Name the faces, edges, and vertices for a sphere.
Why does the formula have r^3 instead of r^2 x h?
Find the volume of the part of the glass that holds water.
Find the radius of a sphere whose volume = 43π m^3.
If we triple the radius of the sphere, how will that change the volume of the sphere?
Find the volume of the sphere if the volume of a cone with the same radius and h = 2r is 297 in^3 .
What do I need help with?
Unit 7 Math Practice Links
Live Lesson Recording
Function of the Day:
How would we find the volume of the pencil? What else would we need to know?
What do you know?
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Vocabulary Terms - Define/Illustrate
Describe the differences between a Sphere, a Cone, and a Cylinder?
How are the formulas for these circular solids related to each other?
5 Problems - SHOW ALL WORK!
Name the faces, edges, and vertices for a sphere.
Why does the formula have r^3 instead of r^2 x h?
Find the volume of the part of the glass that holds water.
Find the radius of a sphere whose volume = 43π m^3.
If we triple the radius of the sphere, how will that change the volume of the sphere?
Find the volume of the sphere if the volume of a cone with the same radius and h = 2r is 297 in^3 .
What do I need help with?
Answers
GPT-4o mini
**Function of the Day:**
To find the volume of a pencil, we would typically model it as a geometric shape. Most often, a pencil can be approximated as a cylinder (for the body) and a cone (for the tip). We would need to know the following:
1. The radius of the cylindrical body of the pencil.
2. The height (length) of the cylindrical part.
3. The radius of the conical tip.
4. The height of the conical tip.
**What do you know?**
1. The volume of a cylinder is given by the formula \( V = \pi r^2 h \).
2. The volume of a cone is given by the formula \( V = \frac{1}{3} \pi r^2 h \).
3. The volume of a sphere is given by the formula \( V = \frac{4}{3} \pi r^3 \).
4. A sphere has no edges or vertices, while a cone has 1 vertex and 1 circular face, and a cylinder has 2 circular faces and no vertices.
5. To find the volume of complex shapes, we may need to break them into simpler shapes, calculate their volumes separately, and then add them together.
6. The dimension \( r^3 \) appears in the sphere's volume formula because it involves three dimensions — radius for width, height, and depth.
7. The volume formulas for circular solids involve \(\pi\) because they relate to circles, which are foundational to these three-dimensional shapes.
8. The concept of volume involves calculating how much space an object occupies.
9. Understanding geometric solids helps in real-world applications such as engineering and architecture.
10. It’s important to accurately measure dimensions to get correct volume calculations.
**Vocabulary Terms - Define/Illustrate**
1. **Sphere**: A perfectly round three-dimensional object where every point on the surface is equidistant from the center. (Illustrate with a basketball or a globe)
2. **Cone**: A three-dimensional shape with a circular base that tapers smoothly from the base to a single vertex (the tip). (Illustrate with an ice cream cone)
3. **Cylinder**: A three-dimensional shape with two parallel circular bases connected by a curved surface. (Illustrate with a soup can)
**Describe the differences between a Sphere, a Cone, and a Cylinder?**
- **Shape**: A sphere has a continuous curved surface with no edges or vertices. A cone has a circular base and narrows to a point (vertex). A cylinder has two circular bases connected by straight sides.
- **Edges & Vertices**: A sphere has none, a cone has one edge (the circular base) and one vertex (the tip), while a cylinder has two circular edges (bases) and no vertices.
- **Volume Calculation**: The formulas differ based on their shape and dimensions; the sphere involves cubing the radius (for three dimensions), the cone integrates the height, and the cylinder involves multiplying the area of the circular base by the height.
**How are the formulas for these circular solids related to each other?**
The formula for the volume of a cone is derived from the cylinder — specifically, a cone’s volume is one-third that of a cylinder with the same base and height. The sphere’s formula incorporates volume proportional to the radius cubed, indicating that it fills space in three dimensions.
---
**5 Problems - SHOW ALL WORK!**
1. **Name the faces, edges, and vertices for a sphere.**
- **Faces**: 1 (the curved surface)
- **Edges**: 0
- **Vertices**: 0
2. **Why does the formula have \( r^3 \) instead of \( r^2 \times h \)?**
- The formula for the volume of a sphere, \( V = \frac{4}{3} \pi r^3 \), includes \( r^3 \) because it measures the three-dimensional space the sphere occupies, relating to all three dimensions (width, height, and depth).
3. **Find the volume of the part of the glass that holds water.**
- Suppose the glass is in the shape of a cylinder with height \( h \) and radius \( r \):
\[
V = \pi r^2 h
\]
(Insert given values for \( r \) and \( h \) for calculations.)
4. **Find the radius of a sphere whose volume = \( 43\pi m^3 \).**
- Using the volume formula:
\[
V = \frac{4}{3} \pi r^3 = 43 \pi
\]
We cancel \( \pi \) and solve for \( r \):
\[
\frac{4}{3} r^3 = 43
\]
\[
r^3 = \frac{43 \times 3}{4} = \frac{129}{4} = 32.25
\]
\[
r = \sqrt[3]{32.25} \approx 3.18 \text{ m}
\]
5. **If we triple the radius of the sphere, how will that change the volume of the sphere?**
- Let the original radius be \( r \). The new radius will be \( 3r \).
- The new volume will be:
\[
V = \frac{4}{3} \pi (3r)^3 = \frac{4}{3} \pi \times 27r^3 = 27 \left( \frac{4}{3} \pi r^3 \right)
\]
- Hence, tripling the radius increases the volume by a factor of \( 27 \).
6. **Find the volume of the sphere if the volume of a cone with the same radius and \( h = 2r \) is \( 297 \, in^3 \).**
- Volume of the cone:
\[
V = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3
\]
- Set equal to \( 297 \):
\[
\frac{2}{3} \pi r^3 = 297
\]
\[
r^3 = \frac{297 \times 3}{2\pi} \approx \frac{891}{6.2832} \approx 141.32
\]
- Thus, the volume of the sphere:
\[
V_{sphere} = \frac{4}{3} \pi r^3 = \frac{4}{3} \pi (141.32) \approx \frac{4 \times 3.1416 \times 141.32}{3} \approx 592.26 \, in^3
\]
**What do I need help with?**
- If there are specific concepts or equations that are unclear, identify them.
- If you're unsure about how to apply formulas to real-life problems, articulate those questions for further clarification.
To find the volume of a pencil, we would typically model it as a geometric shape. Most often, a pencil can be approximated as a cylinder (for the body) and a cone (for the tip). We would need to know the following:
1. The radius of the cylindrical body of the pencil.
2. The height (length) of the cylindrical part.
3. The radius of the conical tip.
4. The height of the conical tip.
**What do you know?**
1. The volume of a cylinder is given by the formula \( V = \pi r^2 h \).
2. The volume of a cone is given by the formula \( V = \frac{1}{3} \pi r^2 h \).
3. The volume of a sphere is given by the formula \( V = \frac{4}{3} \pi r^3 \).
4. A sphere has no edges or vertices, while a cone has 1 vertex and 1 circular face, and a cylinder has 2 circular faces and no vertices.
5. To find the volume of complex shapes, we may need to break them into simpler shapes, calculate their volumes separately, and then add them together.
6. The dimension \( r^3 \) appears in the sphere's volume formula because it involves three dimensions — radius for width, height, and depth.
7. The volume formulas for circular solids involve \(\pi\) because they relate to circles, which are foundational to these three-dimensional shapes.
8. The concept of volume involves calculating how much space an object occupies.
9. Understanding geometric solids helps in real-world applications such as engineering and architecture.
10. It’s important to accurately measure dimensions to get correct volume calculations.
**Vocabulary Terms - Define/Illustrate**
1. **Sphere**: A perfectly round three-dimensional object where every point on the surface is equidistant from the center. (Illustrate with a basketball or a globe)
2. **Cone**: A three-dimensional shape with a circular base that tapers smoothly from the base to a single vertex (the tip). (Illustrate with an ice cream cone)
3. **Cylinder**: A three-dimensional shape with two parallel circular bases connected by a curved surface. (Illustrate with a soup can)
**Describe the differences between a Sphere, a Cone, and a Cylinder?**
- **Shape**: A sphere has a continuous curved surface with no edges or vertices. A cone has a circular base and narrows to a point (vertex). A cylinder has two circular bases connected by straight sides.
- **Edges & Vertices**: A sphere has none, a cone has one edge (the circular base) and one vertex (the tip), while a cylinder has two circular edges (bases) and no vertices.
- **Volume Calculation**: The formulas differ based on their shape and dimensions; the sphere involves cubing the radius (for three dimensions), the cone integrates the height, and the cylinder involves multiplying the area of the circular base by the height.
**How are the formulas for these circular solids related to each other?**
The formula for the volume of a cone is derived from the cylinder — specifically, a cone’s volume is one-third that of a cylinder with the same base and height. The sphere’s formula incorporates volume proportional to the radius cubed, indicating that it fills space in three dimensions.
---
**5 Problems - SHOW ALL WORK!**
1. **Name the faces, edges, and vertices for a sphere.**
- **Faces**: 1 (the curved surface)
- **Edges**: 0
- **Vertices**: 0
2. **Why does the formula have \( r^3 \) instead of \( r^2 \times h \)?**
- The formula for the volume of a sphere, \( V = \frac{4}{3} \pi r^3 \), includes \( r^3 \) because it measures the three-dimensional space the sphere occupies, relating to all three dimensions (width, height, and depth).
3. **Find the volume of the part of the glass that holds water.**
- Suppose the glass is in the shape of a cylinder with height \( h \) and radius \( r \):
\[
V = \pi r^2 h
\]
(Insert given values for \( r \) and \( h \) for calculations.)
4. **Find the radius of a sphere whose volume = \( 43\pi m^3 \).**
- Using the volume formula:
\[
V = \frac{4}{3} \pi r^3 = 43 \pi
\]
We cancel \( \pi \) and solve for \( r \):
\[
\frac{4}{3} r^3 = 43
\]
\[
r^3 = \frac{43 \times 3}{4} = \frac{129}{4} = 32.25
\]
\[
r = \sqrt[3]{32.25} \approx 3.18 \text{ m}
\]
5. **If we triple the radius of the sphere, how will that change the volume of the sphere?**
- Let the original radius be \( r \). The new radius will be \( 3r \).
- The new volume will be:
\[
V = \frac{4}{3} \pi (3r)^3 = \frac{4}{3} \pi \times 27r^3 = 27 \left( \frac{4}{3} \pi r^3 \right)
\]
- Hence, tripling the radius increases the volume by a factor of \( 27 \).
6. **Find the volume of the sphere if the volume of a cone with the same radius and \( h = 2r \) is \( 297 \, in^3 \).**
- Volume of the cone:
\[
V = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3
\]
- Set equal to \( 297 \):
\[
\frac{2}{3} \pi r^3 = 297
\]
\[
r^3 = \frac{297 \times 3}{2\pi} \approx \frac{891}{6.2832} \approx 141.32
\]
- Thus, the volume of the sphere:
\[
V_{sphere} = \frac{4}{3} \pi r^3 = \frac{4}{3} \pi (141.32) \approx \frac{4 \times 3.1416 \times 141.32}{3} \approx 592.26 \, in^3
\]
**What do I need help with?**
- If there are specific concepts or equations that are unclear, identify them.
- If you're unsure about how to apply formulas to real-life problems, articulate those questions for further clarification.