Question 1 (Multiple Choice Worth 4 points)

A square prism and a cylinder have the same height. The area of the cross-section of the square prism is 314 square units. Based on this information, which argument can be made?

A. The volume of the square prism is [tex]$\frac{1}{3}$[/tex] the volume of the cylinder.
B. The volume of the square prism is half the volume of the cylinder.
C. The volume of the square prism is equal to the volume of the cylinder.
D. The volume of the square prism is twice the volume of the cylinder.



Answer :

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.