To determine which argument can be made about the volumes of a square prism and a cylinder with the same height, given their cross-sectional areas, let's follow a step-by-step approach.
1. Identify Cross-Sectional Areas:
- The area of the cross-section of the square prism is 628 square units.
- The area of the cross-section of the cylinder is [tex]\( 200\sqrt{3} \)[/tex] square units, which numerically evaluates to approximately 346.41016151377545 square units.
2. Volume Calculation:
- The volume of a prism or cylinder is given by the product of the cross-sectional area and the height.
- Since both shapes have the same height [tex]\( h \)[/tex], their volumes can be compared by comparing their cross-sectional areas.
3. Ratio of Volumes:
- The volume of the square prism [tex]\( V_{\text{square prism}} \)[/tex] is proportional to its cross-sectional area, hence:
[tex]\[ V_{\text{square prism}} \propto 628 \][/tex]
- The volume of the cylinder [tex]\( V_{\text{cylinder}} \)[/tex] is proportional to its cross-sectional area, hence:
[tex]\[ V_{\text{cylinder}} \propto 200\sqrt{3} \approx 346.41016151377545 \][/tex]
4. Calculate the Ratio:
- The ratio of the volumes will be the same as the ratio of their cross-sectional areas since the height [tex]\( h \)[/tex] is constant and will cancel out in the ratio.
[tex]\[ \text{Ratio} = \frac{V_{\text{square prism}}}{V_{\text{cylinder}}} = \frac{628}{200\sqrt{3}} \approx \frac{628}{346.41016151377545} \approx 1.812879845255425 \][/tex]
5. Interpret the Ratio:
- The ratio [tex]\( \approx 1.812879845255425 \)[/tex] indicates that the volume of the square prism is approximately 1.813 times the volume of the cylinder.
Therefore, the argument that can be made based on this information is that the volume of the square prism is nearly 1.813 times the volume of the cylinder. Since this does not exactly match any of the provided options directly, we can conclude:
- The volume of the square prism is more than the volume of the cylinder but less than twice the volume of the cylinder.
Thus, the closest correct interpretation might be understood as that the volume of the square prism is (practically) larger than the volume of the cylinder, approximately 1.813 times. This does not match exactly any given multiple-choice options but best fits the context provided in the problem.
