Question
Two stacks of 23 quarters each are shown below. One stack forms a cylinder but the other stack does not form a cylinder. Use Cavalieri's Principle to explain why the volumes of these two stacks of quarters are equal. Give your answers in complete sentences and show your work for any formulas or calculations needed.
Answers
GPT-4o mini
Cavalieri's Principle states that if two solids of equal height have the same cross-sectional areas at every level, then the volumes of those solids are equal, regardless of their shapes. In this case, we have two stacks of 23 quarters each – one stack arranged as a cylinder and the other stack in a different arrangement that does not form a cylinder.
Let's analyze both stacks:
1. **Volume of the cylindrical stack:**
The volume \(V\) of a cylinder can be calculated using the formula:
\[
V = \pi r^2 h
\]
where \(r\) is the radius of the base and \(h\) is the height.
For the stack arranged as a cylinder, we can consider the cylindrical structure and treat the quarters as uniform discs. Since there are 23 quarters stacked directly on top of each other, the height \(h\) of the cylinder can be calculated as the thickness of one quarter multiplied by 23. Let's assume the thickness of a quarter is approximately 1.75 mm (which is about 0.00175 m).
Thus, the height \(h\) is:
\[
h = 23 \times 0.00175 \text{ m} = 0.04025 \text{ m}
\]
To form a cylinder, all cross-sections perpendicular to the height will be circles of equal radius (based on how the quarters are stacked), and thus the cross-sectional area remains constant.
2. **Volume of the non-cylindrical stack:**
For the stack that does not form a perfect cylinder, the shape may be irregular but consists of the same number of quarters (23) stacked upon each other. Regardless of how they are arranged, as long as the total height is the same and the cross-sectional area through any horizontal slice is equivalent and composed solely of quarters, we can affirm the volume remains the same.
In this scenario, if we take a horizontal cross-section at any height, it will still contain the same total area occupied by the 23 quarters, which totals the same area as those in a cylinder. Thus, at every height of this structure, the total cross-sectional area can be calculated in the same manner.
Conclusively, according to Cavalieri's Principle, the volume of both stacks is equal because they both occupy the same height and consist of the same number of identical quarters that maintain a constant cross-sectional area, despite the shape difference. Therefore, even though one stack is cylindrical and the other is not, both stacks have equal volumes.
In summary, the volumes of the two stacks of quarters are equal because, despite their shape differences, they both exhibit the same height and consistent cross-sectional area at every level, fulfilling the criteria set by Cavalieri's Principle.
Let's analyze both stacks:
1. **Volume of the cylindrical stack:**
The volume \(V\) of a cylinder can be calculated using the formula:
\[
V = \pi r^2 h
\]
where \(r\) is the radius of the base and \(h\) is the height.
For the stack arranged as a cylinder, we can consider the cylindrical structure and treat the quarters as uniform discs. Since there are 23 quarters stacked directly on top of each other, the height \(h\) of the cylinder can be calculated as the thickness of one quarter multiplied by 23. Let's assume the thickness of a quarter is approximately 1.75 mm (which is about 0.00175 m).
Thus, the height \(h\) is:
\[
h = 23 \times 0.00175 \text{ m} = 0.04025 \text{ m}
\]
To form a cylinder, all cross-sections perpendicular to the height will be circles of equal radius (based on how the quarters are stacked), and thus the cross-sectional area remains constant.
2. **Volume of the non-cylindrical stack:**
For the stack that does not form a perfect cylinder, the shape may be irregular but consists of the same number of quarters (23) stacked upon each other. Regardless of how they are arranged, as long as the total height is the same and the cross-sectional area through any horizontal slice is equivalent and composed solely of quarters, we can affirm the volume remains the same.
In this scenario, if we take a horizontal cross-section at any height, it will still contain the same total area occupied by the 23 quarters, which totals the same area as those in a cylinder. Thus, at every height of this structure, the total cross-sectional area can be calculated in the same manner.
Conclusively, according to Cavalieri's Principle, the volume of both stacks is equal because they both occupy the same height and consist of the same number of identical quarters that maintain a constant cross-sectional area, despite the shape difference. Therefore, even though one stack is cylindrical and the other is not, both stacks have equal volumes.
In summary, the volumes of the two stacks of quarters are equal because, despite their shape differences, they both exhibit the same height and consistent cross-sectional area at every level, fulfilling the criteria set by Cavalieri's Principle.