Question
Math Practice U7 L5
Unit 7 Math Practice Links - Click here
Live Lesson Recording
*Passcode can be found on the Connexus Pushpin under your Math Class! Use the one for Mrs. Garcia!
Function of the Day:
These 2 cylinders have the same volume.
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
Volume vs Surface Area
Perpendicular applied to height
Keeping π in the answer or not:
5 Problems - SHOW ALL WORK!
Find the volume of the part of the glass that holds the water.
Find the volume of the cylinder:
Find the volume of the soda cans:
Find the volume of the roll of toilet paper.
What common mistakes could you make when finding the volume of a cylinder using the formula: V = r2h?
What do I need help with?
Unit 7 Math Practice Links - Click here
Live Lesson Recording
*Passcode can be found on the Connexus Pushpin under your Math Class! Use the one for Mrs. Garcia!
Function of the Day:
These 2 cylinders have the same volume.
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
Volume vs Surface Area
Perpendicular applied to height
Keeping π in the answer or not:
5 Problems - SHOW ALL WORK!
Find the volume of the part of the glass that holds the water.
Find the volume of the cylinder:
Find the volume of the soda cans:
Find the volume of the roll of toilet paper.
What common mistakes could you make when finding the volume of a cylinder using the formula: V = r2h?
What do I need help with?
Answers
GPT-4o mini
Let’s break down this math practice and address each part.
## Function of the Day:
**These 2 cylinders have the same volume. Can this be true? How could this happen?**
Yes, two cylinders can have the same volume if they have different dimensions. Volume \( V \) of a cylinder is given by the formula \( V = \pi r^2 h \), where \( r \) is the radius and \( h \) is the height. For example, a cylinder with a larger radius can have a shorter height, and another with a smaller radius can have a taller height, yet both can yield the same volume.
## What do you know?
1. The formula for the volume of a cylinder is \( V = \pi r^2 h \).
2. The volume is measured in cubic units (e.g., cubic centimeters).
3. Two cylinders can have the same volume with different dimensions.
4. The relationship between the radius and height affects the total volume.
5. Surface area is a different measurement than volume.
6. The constant \( \pi \) is approximately equal to 3.14 or can be left in terms of \( \pi \).
7. The unit for surface area is square units (e.g., square centimeters).
8. Perpendicular relates to the height being at a right angle to the base of the cylinder.
9. Misusing the units can lead to erroneous volume calculations.
10. It's important to use consistent units (all in cm, m, etc.) when calculating volume.
## Vocabulary Terms - Define/Illustrate
- **Volume vs Surface Area**:
- **Volume**: Measures the space within a 3D object, calculated for cylinders as \( V = \pi r^2 h \).
- **Surface Area**: Measures the total area of the surface of a 3D object, for cylinders as \( SA = 2\pi r(h + r) \).
- **Perpendicular applied to height**: The height of the cylinder is the straight distance from the base to the top, and it forms a right angle (90 degrees) with the circular base.
- **Keeping π in the answer or not**: Sometimes it’s preferred to leave \(\pi\) in the answer for exact calculations, while other times it might be rounded for practical applications.
## 5 Problems - SHOW ALL WORK!
1. **Find the volume of the part of the glass that holds the water.**
- **Example:**
If the glass is a cylinder with a radius of 3 cm and a height of 10 cm:
\[
V = \pi r^2 h = \pi (3)^2 (10) = 90\pi \, \text{cm}^3 \approx 282.74 \, \text{cm}^3.
\]
2. **Find the volume of the cylinder:**
- **Example:**
Radius = 5 cm, Height = 12 cm:
\[
V = \pi (5)^2 (12) = 300\pi \, \text{cm}^3 \approx 942.48 \, \text{cm}^3.
\]
3. **Find the volume of the soda cans:**
- **Example:**
Radius = 6 cm, Height = 15 cm:
\[
V = \pi (6)^2 (15) = 540\pi \, \text{cm}^3 \approx 1698.46 \, \text{cm}^3.
\]
4. **Find the volume of the roll of toilet paper:**
- **Example:**
Outer radius = 6.5 cm, Inner radius = 4.5 cm, Height = 10 cm (use the outer cylinder volume minus the inner cylinder volume):
\[
V = \pi (6.5)^2 (10) - \pi (4.5)^2 (10) = (42.25\pi - 20.25\pi)(10) = 220\pi \, \text{cm}^3 \approx 691.15 \, \text{cm}^3.
\]
5. **What common mistakes could you make when finding the volume of a cylinder using the formula: V = r²h?**
- Forgetting to multiply by \(\pi\).
- Using the wrong units (cm instead of m or vice versa).
- Failing to square the radius (calculating \( r \times h\) instead of \( r^2 \)).
- Misplacing the values or substituting incorrectly.
## What do I need help with?
Feel free to specify any areas or problems that you find challenging, and I can provide targeted assistance!
## Function of the Day:
**These 2 cylinders have the same volume. Can this be true? How could this happen?**
Yes, two cylinders can have the same volume if they have different dimensions. Volume \( V \) of a cylinder is given by the formula \( V = \pi r^2 h \), where \( r \) is the radius and \( h \) is the height. For example, a cylinder with a larger radius can have a shorter height, and another with a smaller radius can have a taller height, yet both can yield the same volume.
## What do you know?
1. The formula for the volume of a cylinder is \( V = \pi r^2 h \).
2. The volume is measured in cubic units (e.g., cubic centimeters).
3. Two cylinders can have the same volume with different dimensions.
4. The relationship between the radius and height affects the total volume.
5. Surface area is a different measurement than volume.
6. The constant \( \pi \) is approximately equal to 3.14 or can be left in terms of \( \pi \).
7. The unit for surface area is square units (e.g., square centimeters).
8. Perpendicular relates to the height being at a right angle to the base of the cylinder.
9. Misusing the units can lead to erroneous volume calculations.
10. It's important to use consistent units (all in cm, m, etc.) when calculating volume.
## Vocabulary Terms - Define/Illustrate
- **Volume vs Surface Area**:
- **Volume**: Measures the space within a 3D object, calculated for cylinders as \( V = \pi r^2 h \).
- **Surface Area**: Measures the total area of the surface of a 3D object, for cylinders as \( SA = 2\pi r(h + r) \).
- **Perpendicular applied to height**: The height of the cylinder is the straight distance from the base to the top, and it forms a right angle (90 degrees) with the circular base.
- **Keeping π in the answer or not**: Sometimes it’s preferred to leave \(\pi\) in the answer for exact calculations, while other times it might be rounded for practical applications.
## 5 Problems - SHOW ALL WORK!
1. **Find the volume of the part of the glass that holds the water.**
- **Example:**
If the glass is a cylinder with a radius of 3 cm and a height of 10 cm:
\[
V = \pi r^2 h = \pi (3)^2 (10) = 90\pi \, \text{cm}^3 \approx 282.74 \, \text{cm}^3.
\]
2. **Find the volume of the cylinder:**
- **Example:**
Radius = 5 cm, Height = 12 cm:
\[
V = \pi (5)^2 (12) = 300\pi \, \text{cm}^3 \approx 942.48 \, \text{cm}^3.
\]
3. **Find the volume of the soda cans:**
- **Example:**
Radius = 6 cm, Height = 15 cm:
\[
V = \pi (6)^2 (15) = 540\pi \, \text{cm}^3 \approx 1698.46 \, \text{cm}^3.
\]
4. **Find the volume of the roll of toilet paper:**
- **Example:**
Outer radius = 6.5 cm, Inner radius = 4.5 cm, Height = 10 cm (use the outer cylinder volume minus the inner cylinder volume):
\[
V = \pi (6.5)^2 (10) - \pi (4.5)^2 (10) = (42.25\pi - 20.25\pi)(10) = 220\pi \, \text{cm}^3 \approx 691.15 \, \text{cm}^3.
\]
5. **What common mistakes could you make when finding the volume of a cylinder using the formula: V = r²h?**
- Forgetting to multiply by \(\pi\).
- Using the wrong units (cm instead of m or vice versa).
- Failing to square the radius (calculating \( r \times h\) instead of \( r^2 \)).
- Misplacing the values or substituting incorrectly.
## What do I need help with?
Feel free to specify any areas or problems that you find challenging, and I can provide targeted assistance!