To compare the square prism and the cylinder based on the information provided, let's analyze the data:
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Cross-sectional Area of the Square Prism: The area of the cross-section of the square prism is given as \(314\) square units. Since the cross-section is square in shape, we can find the side length of the prism using the formula for the area of a square, \(A = s^2\), where \(s\) is the side length.
\[ s^2 = 314 \implies s = \sqrt{314} \]
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Cross-sectional Area of the Cylinder: The area of the circular cross-section of the cylinder is given as \(50\pi\) square units. The area of a circle is calculated using the formula \(A = \pi r^2\), where \(r\) is the radius.
\[ \pi r^2 = 50\pi \implies r^2 = 50 \implies r = \sqrt{50} = 5\sqrt{2} \]
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Now that we have the necessary dimensions, we can argue about the respective shapes based on the relationships of their dimensions, particularly focusing on their heights since they are the same:
- The square prism has a height that can be determined if we use the cross-sectional area and volume formulas, but we do not need the volume to make a comparison in this case since both have the same height.
- The cylinder's cross-section doesn't imply further comparison, but its radius is \(5\sqrt{2}\).
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Comparison of Areas: Since both are prisms with the same height, the volumes will be a direct comparison of their cross-sectional areas multiplied by their common height. Since \(314 > 50\pi\) (where \(50\pi \approx 157\)), we can conclude that in terms of area and thus volume: \[ 314 \text{ square units > } 50\pi \text{ square units (approximately 157 square units)} \] This implies that the square prism has a larger cross-sectional area than the cylinder.
Conclusion: The argument we can make here is that for any fixed height, the square prism will have a greater volume than the cylinder because its cross-sectional area is larger, despite both having the same height.