Question
Math Practice U7 L6
Unit 7 Math Practice Links - Click here
Function of the Day:
The volume of the cone and the cylinder are the same.
Can this be true? How could this happen?
What do you know?
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Vocabulary Terms - Define/Illustrate
Difference between a Cylinder and a Cone:
When to use Pythagorean Theorem and when to use volume?
Examples of Cones in real-life:
5 Problems - SHOW ALL WORK!
Find the volume of the part of the glass that holds water.
How many ounces of tea will fill the glass in #1 if
1 cm^3 = 0.0338 ounces?
If we double the height of the glass, how would this affect the volume in cm? Why?
If we double the radius of the glass, how would this affect the volume in cm? Why?
How many cubic ft of grain can the silo hold?
What do I need help with?
Unit 7 Math Practice Links - Click here
Function of the Day:
The volume of the cone and the cylinder are the same.
Can this be true? How could this happen?
What do you know?
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Vocabulary Terms - Define/Illustrate
Difference between a Cylinder and a Cone:
When to use Pythagorean Theorem and when to use volume?
Examples of Cones in real-life:
5 Problems - SHOW ALL WORK!
Find the volume of the part of the glass that holds water.
How many ounces of tea will fill the glass in #1 if
1 cm^3 = 0.0338 ounces?
If we double the height of the glass, how would this affect the volume in cm? Why?
If we double the radius of the glass, how would this affect the volume in cm? Why?
How many cubic ft of grain can the silo hold?
What do I need help with?
Answers
GPT-4o mini
### Math Practice U7 L6
**Function of the Day:**
The statement "The volume of the cone and the cylinder are the same" can be true under specific conditions. Specifically, the volume of a cone is one-third the volume of a cylinder with the same base radius and height. Therefore, for the volumes to be equal, the height of the cone must be three times the height of the cylinder, and both must share the same radius.
---
### What do you know?
1. The formula for the volume of a cylinder is \( V = \pi r^2 h \).
2. The formula for the volume of a cone is \( V = \frac{1}{3} \pi r^2 h \).
3. The height and radius are critical variables in determining the volumes of both shapes.
4. Knowledge of the relationship between the cone's volume and cylinder's volume is useful in solving volume problems.
5. Volume can be compared using units (cubic centimeters, cubic meters, etc.).
6. When the dimensions of a shape are changed, the volume is affected based on the geometric properties.
7. The Pythagorean theorem is often used in problems involving right triangles, while volume formulas are specific to 3D shapes.
8. Real-life examples of cones include ice cream cones, traffic cones, and funnels.
9. Real-life examples of cylinders include cans, tubes, and silos.
10. Doubling dimensions affects volume based on the exponent of the dimensional change.
---
### Vocabulary Terms - Define/Illustrate
**Difference between a Cylinder and a Cone:**
- **Cylinder:** A 3D shape with two parallel circular bases and a uniform height. Volume is calculated using radius and height.
- **Cone:** A 3D shape with a circular base that narrows to a point called the apex. Volume is calculated based on the base radius and height, with the volume being one-third that of a cylinder with the same base radius and height.
---
### When to use Pythagorean Theorem and when to use volume?
- **Pythagorean Theorem:** Used when dealing with right triangles to find a missing side length.
- **Volume:** Used when calculating capacity or space inside a 3D shape using appropriate volume formulas.
---
### Examples of Cones in real-life:
- Ice cream cones
- Christmas trees
- Funnels
- Conical hats
---
### 5 Problems - SHOW ALL WORK!
1. **Find the volume of the part of the glass that holds water.**
Assume the glass has a cone shape with radius \( r \) cm and height \( h \) cm.
\[
V = \frac{1}{3} \pi r^2 h
\]
2. **How many ounces of tea will fill the glass in #1 if** \( 1 \text{ cm}^3 = 0.0338 \text{ ounces}? \)
Calculate the volume in cubic cm from #1 and then convert it to ounces.
\[
\text{Volume in ounces} = V \times 0.0338
\]
3. **If we double the height of the glass, how would this affect the volume in cm? Why?**
Let \( h' = 2h \). The new volume \( V' \) would be:
\[
V' = \frac{1}{3} \pi r^2 (2h) = 2 \left(\frac{1}{3} \pi r^2 h\right) = 2V
\]
Doubling the height doubles the volume.
4. **If we double the radius of the glass, how would this affect the volume in cm? Why?**
Let \( r' = 2r \). The new volume \( V'' \) would be:
\[
V'' = \frac{1}{3} \pi (2r)^2 h = \frac{1}{3} \pi (4r^2)h = 4 \left(\frac{1}{3} \pi r^2 h\right) = 4V
\]
Doubling the radius quadruples the volume.
5. **How many cubic ft of grain can the silo hold?**
Assuming the silo is a cylinder with radius \( r \) ft and height \( h \) ft, we calculate:
\[
V = \pi r^2 h
\]
---
### What do I need help with?
List any specific concepts, problems, or types of equations you find challenging or want clarification on.
**Function of the Day:**
The statement "The volume of the cone and the cylinder are the same" can be true under specific conditions. Specifically, the volume of a cone is one-third the volume of a cylinder with the same base radius and height. Therefore, for the volumes to be equal, the height of the cone must be three times the height of the cylinder, and both must share the same radius.
---
### What do you know?
1. The formula for the volume of a cylinder is \( V = \pi r^2 h \).
2. The formula for the volume of a cone is \( V = \frac{1}{3} \pi r^2 h \).
3. The height and radius are critical variables in determining the volumes of both shapes.
4. Knowledge of the relationship between the cone's volume and cylinder's volume is useful in solving volume problems.
5. Volume can be compared using units (cubic centimeters, cubic meters, etc.).
6. When the dimensions of a shape are changed, the volume is affected based on the geometric properties.
7. The Pythagorean theorem is often used in problems involving right triangles, while volume formulas are specific to 3D shapes.
8. Real-life examples of cones include ice cream cones, traffic cones, and funnels.
9. Real-life examples of cylinders include cans, tubes, and silos.
10. Doubling dimensions affects volume based on the exponent of the dimensional change.
---
### Vocabulary Terms - Define/Illustrate
**Difference between a Cylinder and a Cone:**
- **Cylinder:** A 3D shape with two parallel circular bases and a uniform height. Volume is calculated using radius and height.
- **Cone:** A 3D shape with a circular base that narrows to a point called the apex. Volume is calculated based on the base radius and height, with the volume being one-third that of a cylinder with the same base radius and height.
---
### When to use Pythagorean Theorem and when to use volume?
- **Pythagorean Theorem:** Used when dealing with right triangles to find a missing side length.
- **Volume:** Used when calculating capacity or space inside a 3D shape using appropriate volume formulas.
---
### Examples of Cones in real-life:
- Ice cream cones
- Christmas trees
- Funnels
- Conical hats
---
### 5 Problems - SHOW ALL WORK!
1. **Find the volume of the part of the glass that holds water.**
Assume the glass has a cone shape with radius \( r \) cm and height \( h \) cm.
\[
V = \frac{1}{3} \pi r^2 h
\]
2. **How many ounces of tea will fill the glass in #1 if** \( 1 \text{ cm}^3 = 0.0338 \text{ ounces}? \)
Calculate the volume in cubic cm from #1 and then convert it to ounces.
\[
\text{Volume in ounces} = V \times 0.0338
\]
3. **If we double the height of the glass, how would this affect the volume in cm? Why?**
Let \( h' = 2h \). The new volume \( V' \) would be:
\[
V' = \frac{1}{3} \pi r^2 (2h) = 2 \left(\frac{1}{3} \pi r^2 h\right) = 2V
\]
Doubling the height doubles the volume.
4. **If we double the radius of the glass, how would this affect the volume in cm? Why?**
Let \( r' = 2r \). The new volume \( V'' \) would be:
\[
V'' = \frac{1}{3} \pi (2r)^2 h = \frac{1}{3} \pi (4r^2)h = 4 \left(\frac{1}{3} \pi r^2 h\right) = 4V
\]
Doubling the radius quadruples the volume.
5. **How many cubic ft of grain can the silo hold?**
Assuming the silo is a cylinder with radius \( r \) ft and height \( h \) ft, we calculate:
\[
V = \pi r^2 h
\]
---
### What do I need help with?
List any specific concepts, problems, or types of equations you find challenging or want clarification on.