Question

question 6 (multiple choice worth 1 points) (05.01 mc) the cross - sectional areas of a triangular prism and a right cylinder are congruent. the triangular prism has a height of 5 units, and the right cylinder has a height of 3 units. which conclusion can be made from the given information?
○ the volume of the prism is half the volume of the cylinder.
○ the volume of the prism is twice the volume of the cylinder.
○ the volume of the prism is equal to the volume of the cylinder.
○ the volume of the prism is not equal to the volume of the cylinder.

Explanation:

Step1: Recall Volume Formulas

The volume of a prism (including triangular prism) is \( V_{\text{prism}} = B \times h_{\text{prism}} \), where \( B \) is the cross - sectional area and \( h_{\text{prism}} \) is the height of the prism. The volume of a cylinder is \( V_{\text{cylinder}}=B\times h_{\text{cylinder}} \), where \( B \) is the cross - sectional area (area of the base, which is a circle for a cylinder, but in this case, the cross - sectional areas of the prism and cylinder are congruent, so we use the same \( B \)) and \( h_{\text{cylinder}} \) is the height of the cylinder.

Step2: Substitute Given Heights

We know that the cross - sectional area \( B \) of the triangular prism and the right cylinder are congruent. The height of the triangular prism \( h_{\text{prism}} = 5 \) units and the height of the cylinder \( h_{\text{cylinder}}=3 \) units.

Calculate the volume of the prism: \( V_{\text{prism}}=B\times5 = 5B \)

Calculate the volume of the cylinder: \( V_{\text{cylinder}}=B\times3=3B \)

Now, let's check the relationships:

• For the first option: Is \( 5B=\frac{1}{2}\times3B=\frac{3B}{2} \)? \( 5B

eq\frac{3B}{2} \), so this is false.

• For the second option: Is \( 5B = 2\times3B=6B \)? \( 5B

eq6B \), so this is false.

• For the third option: Is \( 5B = 3B \)? \( 5B

eq3B \), so this is false.

• For the fourth option: Since \( 5B

eq3B \), the volume of the prism is not equal to the volume of the cylinder. This is true.

Answer:

The volume of the prism is not equal to the volume of the cylinder. (The option with the text "The volume of the prism is not equal to the volume of the cylinder.")