Question
Here are two formulas for the volume of a cylinder:
"V" equals pi "r" squared times "h"
"V" equals "B" times "h"
Describe how the formulas are related.
Use the sketch tool if it helps you to show your thinking.
"V" equals pi "r" squared times "h"
"V" equals "B" times "h"
Describe how the formulas are related.
Use the sketch tool if it helps you to show your thinking.
Answers
Answered by
GPT-4o mini
The two formulas for the volume of a cylinder relate to each other through the geometric properties of the cylinder itself.
1. **First Formula**: \( V = \pi r^2 h \)
- In this formula, \( V \) represents the volume of the cylinder, \( r \) is the radius of the circular base, and \( h \) is the height of the cylinder.
- The term \( \pi r^2 \) calculates the area of the circular base (denoted as \( B \)), because the area of a circle is given by \( A = \pi r^2 \).
2. **Second Formula**: \( V = B \times h \)
- In this formula, \( B \) is the area of the base of the cylinder, and \( h \) is the height.
- This is a more general formula for finding the volume of any prism, where the volume is the area of the base multiplied by the height.
### Relationship Between the Formulas
- The first formula directly incorporates the area of the base as part of its calculation, while the second formula emphasizes the concept of using the area of the base \( B \) as a separate term.
- For a cylinder, we can substitute \( B \) with \( \pi r^2 \) in the second formula. Thus, we see that both formulas ultimately represent the same concept of volume. This shows that:
\[ V = B \times h \]
where \( B = \pi r^2 \).
### Sketch Tool
If you were to sketch this, you could draw a cylinder and label the following:
- The circular base at the bottom with the radius \( r \).
- Indicate the height \( h \) of the cylinder.
- Show the area of the base as \( B \), where you can fill in the area formula \( \pi r^2 \).
- Optionally, you can annotate the two volume formulas beside the sketch.
This visual representation would help solidify the understanding of how the two formulas connect to the geometry of the cylinder.
1. **First Formula**: \( V = \pi r^2 h \)
- In this formula, \( V \) represents the volume of the cylinder, \( r \) is the radius of the circular base, and \( h \) is the height of the cylinder.
- The term \( \pi r^2 \) calculates the area of the circular base (denoted as \( B \)), because the area of a circle is given by \( A = \pi r^2 \).
2. **Second Formula**: \( V = B \times h \)
- In this formula, \( B \) is the area of the base of the cylinder, and \( h \) is the height.
- This is a more general formula for finding the volume of any prism, where the volume is the area of the base multiplied by the height.
### Relationship Between the Formulas
- The first formula directly incorporates the area of the base as part of its calculation, while the second formula emphasizes the concept of using the area of the base \( B \) as a separate term.
- For a cylinder, we can substitute \( B \) with \( \pi r^2 \) in the second formula. Thus, we see that both formulas ultimately represent the same concept of volume. This shows that:
\[ V = B \times h \]
where \( B = \pi r^2 \).
### Sketch Tool
If you were to sketch this, you could draw a cylinder and label the following:
- The circular base at the bottom with the radius \( r \).
- Indicate the height \( h \) of the cylinder.
- Show the area of the base as \( B \), where you can fill in the area formula \( \pi r^2 \).
- Optionally, you can annotate the two volume formulas beside the sketch.
This visual representation would help solidify the understanding of how the two formulas connect to the geometry of the cylinder.
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