Question
Describe an informal argument for the volume of a cone. if you know the volume of a cylinder, what must you do to find the volume of a cone that has the same height and base radius?
Answers
Answered by
GPT-4o mini
To informally understand the volume of a cone, let's start with what we know about a cylinder. A cylinder with a base radius \( r \) and height \( h \) has a volume calculated by the formula:
\[
V_{\text{cylinder}} = \pi r^2 h
\]
Now, imagine a cone that has the same base radius \( r \) and height \( h \) as this cylinder. You might visualize filling the cylinder with liquid, and then think about how much of that liquid a cone would hold.
One way to compare the two volumes is to notice that a cone can be thought of as being "pointed" at the top, while the cylinder is "full." If we consider how these two shapes relate, we can think about how the cone fits into the cylinder.
### Informal Argument:
1. **Volume Comparison:** When we look at their shapes, a cone occupies only part of the cylinder. The common observation is that you can fit three cones with the same base and height as the cylinder into that cylindrical space. Therefore, it seems reasonable to conclude that the volume of the cone could be one-third that of the cylinder.
2. **Mathematical Confirmation:** To express this idea mathematically, if the volume of the cylinder is \( \pi r^2 h \), then given our observation about the shapes, we might conclude that the volume of a cone with the same base and height should be:
\[
V_{\text{cone}} = \frac{1}{3} V_{\text{cylinder}} = \frac{1}{3} \pi r^2 h
\]
This leads us to the formula for the volume of a cone:
\[
V_{\text{cone}} = \frac{1}{3} \pi r^2 h
\]
### Summary:
So, informally speaking, if you know the volume of a cylinder (which is \( \pi r^2 h \)), to find the volume of a cone with the same height and base radius, you simply divide the volume of the cylinder by 3. This thought process gives a good intuitive grasp of why a cone's volume is one-third that of a cylinder with the same base and height.
\[
V_{\text{cylinder}} = \pi r^2 h
\]
Now, imagine a cone that has the same base radius \( r \) and height \( h \) as this cylinder. You might visualize filling the cylinder with liquid, and then think about how much of that liquid a cone would hold.
One way to compare the two volumes is to notice that a cone can be thought of as being "pointed" at the top, while the cylinder is "full." If we consider how these two shapes relate, we can think about how the cone fits into the cylinder.
### Informal Argument:
1. **Volume Comparison:** When we look at their shapes, a cone occupies only part of the cylinder. The common observation is that you can fit three cones with the same base and height as the cylinder into that cylindrical space. Therefore, it seems reasonable to conclude that the volume of the cone could be one-third that of the cylinder.
2. **Mathematical Confirmation:** To express this idea mathematically, if the volume of the cylinder is \( \pi r^2 h \), then given our observation about the shapes, we might conclude that the volume of a cone with the same base and height should be:
\[
V_{\text{cone}} = \frac{1}{3} V_{\text{cylinder}} = \frac{1}{3} \pi r^2 h
\]
This leads us to the formula for the volume of a cone:
\[
V_{\text{cone}} = \frac{1}{3} \pi r^2 h
\]
### Summary:
So, informally speaking, if you know the volume of a cylinder (which is \( \pi r^2 h \)), to find the volume of a cone with the same height and base radius, you simply divide the volume of the cylinder by 3. This thought process gives a good intuitive grasp of why a cone's volume is one-third that of a cylinder with the same base and height.