To determine which argument can be made about the volumes of the square prism and the cylinder, we need to compare their volumes. Here’s a detailed, step-by-step solution:
1. Determine the volume of the square prism:
- The volume [tex]\( V_{\text{square prism}} \)[/tex] of the square prism is given by the product of the area of its cross-section and its height.
- Given that the area of the cross-section of the square prism is 314 square units, let’s assume the height [tex]\( h \)[/tex] is 1 unit (as the actual value does not impact the ratio).
- Therefore, [tex]\( V_{\text{square prism}} = \text{Area}_{\text{square prism}} \times h = 314 \times 1 = 314 \)[/tex] cubic units.
2. Determine the volume of the cylinder:
- The volume [tex]\( V_{\text{cylinder}} \)[/tex] of the cylinder is given by the product of the area of its cross-section and its height.
- The area of the cross-section of the cylinder is given as [tex]\( 50\pi \)[/tex] square units, with the same assumed height [tex]\( h = 1 \)[/tex] unit.
- Therefore, [tex]\( V_{\text{cylinder}} = \text{Area}_{\text{cylinder}} \times h = 50\pi \times 1 = 50\pi \approx 157.07963267948966 \)[/tex] cubic units.
3. Calculate the volume ratio:
- The ratio of the volume of the square prism to the volume of the cylinder is given by:
[tex]\[
\text{Volume Ratio} = \frac{V_{\text{square prism}}}{V_{\text{cylinder}}} = \frac{314}{50\pi}
\][/tex]
Simplifying further using the approximation [tex]\( \pi \approx 3.141592653589793 \)[/tex]:
[tex]\[
\text{Volume Ratio} \approx \frac{314}{157.07963267948966} \approx 1.9989860852342054
\][/tex]
4. Interpret the volume ratio:
- The ratio of the volumes [tex]\( 1.9989860852342054 \)[/tex] is very close to 2.
- This means that the volume of the square prism is approximately twice the volume of the cylinder.
Therefore, based on the given cross-sectional areas and the volumes derived, the correct argument is:
The volume of the square prism is twice the volume of the cylinder.
