To determine the relationship between the volume of the square prism and the cylinder, follow these steps:
1. Understand the cross-sectional area:
The area of the cross-section of the square prism is given as 314 square units.
2. Volume of the square prism:
The volume of a square prism can be calculated by multiplying the area of its base by its height. Given the cross-sectional area [tex]\( A = 314 \)[/tex] square units, assume the height [tex]\( h = 1 \)[/tex] unit:
[tex]\[
\text{Volume of the square prism} = A \times h = 314 \times 1 = 314 \text{ cubic units}
\][/tex]
3. Volume of the cylinder:
First, calculate the radius of the cylinder’s circular base using its given cross-sectional area. The area [tex]\( A = 314 \)[/tex] square units can be used to find the radius [tex]\( r \)[/tex] using the formula for the area of a circle [tex]\( A = \pi r^2 \)[/tex]:
[tex]\[
\pi r^2 = 314
\][/tex]
Solving for [tex]\( r \)[/tex]:
[tex]\[
r^2 = \frac{314}{\pi}
\][/tex]
[tex]\[
r = \sqrt{\frac{314}{\pi}}
\][/tex]
Next, the volume of the cylinder can be calculated by multiplying the area of its base by its height. Assume the height [tex]\( h = 1 \)[/tex] unit:
[tex]\[
\text{Volume of the cylinder} = \pi r^2 \times h = 314 \text{ cubic units}
\][/tex]
4. Compare the volumes:
Both the square prism and cylinder have the same height, and their volumes are:
[tex]\[
\text{Volume of the square prism} = 314 \text{ cubic units}
\][/tex]
[tex]\[
\text{Volume of the cylinder} = 314 \text{ cubic units}
\][/tex]
Since both volumes are equal, you can conclude:
The volume of the square prism is equal to the volume of the cylinder.
Therefore, the correct argument is:
- The volume of the square prism is equal to the volume of the cylinder.
