Question
12 How Are They Related? Tr sqrt(± ) Here are two formulas for the volume of a cylinder: V=π r^2· h V=B· h Describe how the formulas are related. Use the sketch tool if it helps you to show your thinking. The volume of a cylinder is the product of the base area and the height. sqrt(± ) Share With Class
Answers
GPT-4o mini
The two formulas for the volume of a cylinder, \( V = \pi r^2 \cdot h \) and \( V = B \cdot h \), are related through the concept of the base area. Let's break this down:
1. **Understanding the Components**:
- In the formula \( V = \pi r^2 \cdot h \):
- \( \pi r^2 \) represents the area of the circular base of the cylinder. Here, \( r \) is the radius of the base, and \( \pi \) is a constant that represents the ratio of the circumference of a circle to its diameter.
- \( h \) is the height of the cylinder.
- In the formula \( V = B \cdot h \):
- \( B \) represents the area of the base of the cylinder, which can be any shape (not just a circle).
2. **Relationship Between the Formulas**:
- The formula \( V = \pi r^2 \cdot h \) can be seen as a specific case of \( V = B \cdot h \) where the base \( B \) is specifically the area of a circle (\( \pi r^2 \)).
- Thus, the volume formula \( V = B \cdot h \) is more general and can apply to any prism or cylinder, whereas \( V = \pi r^2 \cdot h \) specifically describes the volume of a cylinder with a circular base.
3. **Visual Representation**:
- You can visualize this relationship by sketching a cylinder.
- Draw a cylinder and label the height \( h \).
- Show a circular base and label the radius \( r \) and the area of the base \( B = \pi r^2 \).
- Indicate that the volume is the base area multiplied by the height (\( V = B \cdot h \)) and that in the case of a circular base, \( B = \pi r^2 \).
In summary, the relationship between the two formulas is that the first formula is a specific instance of the second, where the base \( B \) is the area of a circle. This illustrates a fundamental principle in geometry: when calculating the volume of any solid where the height is perpendicular to the base, the volume can be determined by multiplying the area of the base by the height.
1. **Understanding the Components**:
- In the formula \( V = \pi r^2 \cdot h \):
- \( \pi r^2 \) represents the area of the circular base of the cylinder. Here, \( r \) is the radius of the base, and \( \pi \) is a constant that represents the ratio of the circumference of a circle to its diameter.
- \( h \) is the height of the cylinder.
- In the formula \( V = B \cdot h \):
- \( B \) represents the area of the base of the cylinder, which can be any shape (not just a circle).
2. **Relationship Between the Formulas**:
- The formula \( V = \pi r^2 \cdot h \) can be seen as a specific case of \( V = B \cdot h \) where the base \( B \) is specifically the area of a circle (\( \pi r^2 \)).
- Thus, the volume formula \( V = B \cdot h \) is more general and can apply to any prism or cylinder, whereas \( V = \pi r^2 \cdot h \) specifically describes the volume of a cylinder with a circular base.
3. **Visual Representation**:
- You can visualize this relationship by sketching a cylinder.
- Draw a cylinder and label the height \( h \).
- Show a circular base and label the radius \( r \) and the area of the base \( B = \pi r^2 \).
- Indicate that the volume is the base area multiplied by the height (\( V = B \cdot h \)) and that in the case of a circular base, \( B = \pi r^2 \).
In summary, the relationship between the two formulas is that the first formula is a specific instance of the second, where the base \( B \) is the area of a circle. This illustrates a fundamental principle in geometry: when calculating the volume of any solid where the height is perpendicular to the base, the volume can be determined by multiplying the area of the base by the height.