To solve this problem, we need to calculate the volumes of both prisms and then compare them. Let's go through the calculations step-by-step:
1. Calculate the Volume of Rectangular Prism A:
- The cross-sectional area of prism A is a rectangle with dimensions 1.5 units (width) and 1 unit (height).
- The area of the cross-section can be calculated as:
[tex]\[
\text{Area of cross-section} = \text{width} \times \text{height} = 1.5 \times 1 = 1.5 \text{ square units}
\][/tex]
- The length of prism A is 1.81 units.
- The volume of a rectangular prism is given by:
[tex]\[
\text{Volume}_A = \text{Area of cross-section} \times \text{length} = 1.5 \times 1.81 = 2.715 \text{ cubic units}
\][/tex]
2. Calculate the Volume of Triangular Prism B:
- The cross-sectional area of prism B is a triangle with a base of 2 units and a height of 1.5 units.
- The area of the triangular cross-section can be calculated as:
[tex]\[
\text{Area of triangular cross-section} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 2 \times 1.5 = 1.5 \text{ square units}
\][/tex]
- The length of prism B is 1.81 units.
- The volume of a triangular prism is given by:
[tex]\[
\text{Volume}_B = \text{Area of triangular cross-section} \times \text{length} = 1.5 \times 1.81 = 2.715 \text{ cubic units}
\][/tex]
3. Compare the Volumes:
- We have calculated the volumes as:
[tex]\[
\text{Volume}_A = 2.715 \text{ cubic units}
\][/tex]
[tex]\[
\text{Volume}_B = 2.715 \text{ cubic units}
\][/tex]
- Now let's compare these volumes with the given statements:
- Volume [tex]\( B = \frac{1}{2} \, \text{Volume}_A \)[/tex]:
[tex]\[
\frac{1}{2} \times 2.715 = 1.3575 \, \text{ cubic units}
\][/tex]
- This is not equal to [tex]\( \text{Volume}_B \)[/tex], so this statement is false.
- Volume [tex]\( B = \frac{1}{3} \, \text{Volume}_A \)[/tex]:
[tex]\[
\frac{1}{3} \times 2.715 = 0.905 \, \text{ cubic units}
\][/tex]
- This is not equal to [tex]\( \text{Volume}_B \)[/tex], so this statement is false.
- Volume [tex]\( B = \text{Volume}_A \)[/tex]:
[tex]\[
2.715 = 2.715
\][/tex]
- This statement is true.
- Volume [tex]\( B = 2 \times \text{Volume}_A \)[/tex]:
[tex]\[
2 \times 2.715 = 5.43 \, \text{ cubic units}
\][/tex]
- This is not equal to [tex]\( \text{Volume}_B \)[/tex], so this statement is false.
Thus, the true statement is:
[tex]\[
\text{Volume} \, B = \text{Volume} \, A
\][/tex]
