To determine the correct argument based on the given areas of the cross-sections for the square prism and the cylinder, let's proceed with the following steps:
1. Identify Given Values:
- The area of the cross-section of the square prism, [tex]\(A_{\text{square prism}} = 628\)[/tex] square units.
- The area of the cross-section of the cylinder, [tex]\(A_{\text{cylinder}} = 200\pi\)[/tex] square units.
2. Explain the Relationship:
- Since the height [tex]\(h\)[/tex] of the square prism and the cylinder is the same, the volumes are directly proportional to the areas of their cross-sections. Thus:
[tex]\[
\frac{\text{Volume of Square Prism}}{\text{Volume of Cylinder}} = \frac{A_{\text{square prism}}}{A_{\text{cylinder}}}
\][/tex]
3. Calculate the Ratio:
- Compute the ratio of the areas:
[tex]\[
\text{Ratio} = \frac{A_{\text{square prism}}}{A_{\text{cylinder}}} = \frac{628}{200\pi}
\][/tex]
4. Approximate the Ratio:
- Recall the approximate value of [tex]\(\pi \approx 3.14159\)[/tex], so:
[tex]\[
200\pi \approx 200 \times 3.14159 = 628.318
\][/tex]
- Thus:
[tex]\[
\text{Ratio} = \frac{628}{628.318} \approx 0.99949
\][/tex]
5. Comparison with Options:
- The calculated ratio is approximately [tex]\(0.99949\)[/tex].
- Let's compare this ratio with each given option:
1. The volume of the square prism is [tex]\(\frac{1}{3}\)[/tex] the volume of the cylinder: [tex]\( \frac{1}{3} \approx 0.3333 \)[/tex] (Not close to 0.99949)
2. The volume of the square prism is half the volume of the cylinder: [tex]\( \frac{1}{2} = 0.5 \)[/tex] (Not close to 0.99949)
3. The volume of the square prism is equal to the volume of the cylinder: [tex]\( 1 \)[/tex] (Very close to 0.99949)
4. The volume of the square prism is twice the volume of the cylinder: [tex]\( 2 \)[/tex] (Not close to 0.99949)
6. Conclusion:
- The ratio [tex]\(0.99949\)[/tex] is extremely close to [tex]\(1\)[/tex]. This suggests that the volumes of the square prism and the cylinder are nearly equal.
Given the circumstances and the closely matching value, the correct argument can be concluded as:
None of the given arguments are correct.
