Certainly! Let's go through the process step by step to determine which statement is true about the volumes of the prisms.
1. Calculate the Volume of Rectangular Prism A:
- The cross-sectional dimensions of prism A are 1.5 units (width) and 1 unit (height).
- The length of prism A is 1.81 units.
- To calculate the volume of a rectangular prism, we use the formula:
[tex]\[
\text{Volume of A} = \text{width} \times \text{height} \times \text{length}
\][/tex]
- Substituting in the given values:
[tex]\[
\text{Volume of A} = 1.5 \times 1 \times 1.81 = 2.715 \, \text{cubic units}
\][/tex]
2. Calculate the Volume of Triangular Prism B:
- The base of the triangular cross-section is 2 units, and the height is 1 unit.
- The length of prism B is 1.81 units.
- To calculate the volume of a triangular prism, we use the formula:
[tex]\[
\text{Volume of B} = \frac{1}{2} \times \text{base} \times \text{height} \times \text{length}
\][/tex]
- Substituting in the given values:
[tex]\[
\text{Volume of B} = \frac{1}{2} \times 2 \times 1 \times 1.81 = 1.81 \, \text{cubic units}
\][/tex]
3. Compare the Volumes of Prisms A and B:
- We have [tex]\(\text{Volume of A} = 2.715 \, \text{cubic units}\)[/tex].
- We have [tex]\(\text{Volume of B} = 1.81 \, \text{cubic units}\)[/tex].
4. Calculate the Ratios to Verify Statements:
- To check if [tex]\(\text{Volume B} = \frac{1}{2} \, \text{Volume A}\)[/tex]:
[tex]\[
\frac{1}{2} \, \text{Volume of A} = \frac{1}{2} \times 2.715 = 1.3575 \, \text{cubic units}
\][/tex]
Since 1.81 is not equal to 1.3575, this statement is false.
- To check if [tex]\(\text{Volume B} = \frac{1}{3} \, \text{Volume A}\)[/tex]:
[tex]\[
\frac{1}{3} \, \text{Volume of A} = \frac{1}{3} \times 2.715 = 0.905 \, \text{cubic units}
\][/tex]
Since 1.81 is not equal to 0.905, this statement is false.
- To check if [tex]\(\text{Volume B} = \text{Volume A}\)[/tex]:
[tex]\[
1.81 \, \text{cubic units} \neq 2.715 \, \text{cubic units}
\][/tex]
This statement is false.
- To check if [tex]\(\text{Volume B} = 2 \, \text{Volume A}\)[/tex]:
[tex]\[
2 \times 2.715 = 5.43 \, \text{cubic units}
\][/tex]
Since 1.81 is not equal to 5.43, this statement is false.
Since none of the proposed statements about the relationship between Volume B and Volume A hold, it appears there might be an error in the provided options, or we might need to re-evaluate the options given the correct volumes. Based on the volumes calculated, none of the provided statements is accurate.
