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Sagot :
Let's determine the volumes of two prisms, A and B, and compare them to find out which of the given statements is true.
1. Dimensions and Volumes of Rectangular Prism A:
- Length of A: 1.81 units
- Width of A: 1.5 units (this is one side of the cross-sectional area)
- Height of A: 1 unit (this is the other side of the cross-sectional area)
The formula to calculate the volume of a rectangular prism is given by:
[tex]\[ \text{Volume of A} = \text{length} \times \text{width} \times \text{height} \][/tex]
Substituting the dimensions into the formula:
[tex]\[ \text{Volume of A} = 1.81 \times 1.5 \times 1 = 2.715 \text{ cubic units} \][/tex]
2. Dimensions and Volumes of Triangular Prism B:
- Length of B: 1.81 units
- Base of B: 2 units
- Height of B: 1.5 units (height of the triangular cross-section)
The area of the triangular cross-section is calculated using the formula for the area of a triangle:
[tex]\[ \text{Area of cross-section} = \frac{1}{2} \times \text{base} \times \text{height} \][/tex]
Substituting the values, we get:
[tex]\[ \text{Area of cross-section} = \frac{1}{2} \times 2 \times 1.5 = 1.5 \text{ square units} \][/tex]
Now, compute the volume of the triangular prism using the area of the cross-section:
[tex]\[ \text{Volume of B} = \text{Area of cross-section} \times \text{length} \][/tex]
Substituting the values, we get:
[tex]\[ \text{Volume of B} = 1.5 \times 1.81 = 2.715 \text{ cubic units} \][/tex]
3. Comparing Volumes:
We have:
[tex]\[ \text{Volume of A} = 2.715 \text{ cubic units} \][/tex]
[tex]\[ \text{Volume of B} = 2.715 \text{ cubic units} \][/tex]
Let's compare the volumes to the given statements:
- Volume [tex]\(B = \frac{1}{2} \)[/tex] Volume [tex]\( A \)[/tex]:
[tex]\[ 2.715 = \frac{1}{2} \times 2.715 \quad \text{(not true)} \][/tex]
- Volume [tex]\(B = \frac{1}{3} \)[/tex] Volume [tex]\( A \)[/tex]:
[tex]\[ 2.715 = \frac{1}{3} \times 2.715 \quad \text{(not true)} \][/tex]
- Volume [tex]\(B = \)[/tex] Volume [tex]\( A \)[/tex]:
[tex]\[ 2.715 = 2.715 \quad \text{(true)} \][/tex]
- Volume [tex]\(B = 2 \times \)[/tex] Volume [tex]\( A \)[/tex]:
[tex]\[ 2.715 = 2 \times 2.715 \quad \text{(not true)} \][/tex]
Therefore, the statement that is true is:
[tex]\[ \text{Volume of B} = \text{Volume of A} \][/tex]
So, the correct statement is:
[tex]\[ \text{Volume } B = \text{Volume } A \][/tex]
1. Dimensions and Volumes of Rectangular Prism A:
- Length of A: 1.81 units
- Width of A: 1.5 units (this is one side of the cross-sectional area)
- Height of A: 1 unit (this is the other side of the cross-sectional area)
The formula to calculate the volume of a rectangular prism is given by:
[tex]\[ \text{Volume of A} = \text{length} \times \text{width} \times \text{height} \][/tex]
Substituting the dimensions into the formula:
[tex]\[ \text{Volume of A} = 1.81 \times 1.5 \times 1 = 2.715 \text{ cubic units} \][/tex]
2. Dimensions and Volumes of Triangular Prism B:
- Length of B: 1.81 units
- Base of B: 2 units
- Height of B: 1.5 units (height of the triangular cross-section)
The area of the triangular cross-section is calculated using the formula for the area of a triangle:
[tex]\[ \text{Area of cross-section} = \frac{1}{2} \times \text{base} \times \text{height} \][/tex]
Substituting the values, we get:
[tex]\[ \text{Area of cross-section} = \frac{1}{2} \times 2 \times 1.5 = 1.5 \text{ square units} \][/tex]
Now, compute the volume of the triangular prism using the area of the cross-section:
[tex]\[ \text{Volume of B} = \text{Area of cross-section} \times \text{length} \][/tex]
Substituting the values, we get:
[tex]\[ \text{Volume of B} = 1.5 \times 1.81 = 2.715 \text{ cubic units} \][/tex]
3. Comparing Volumes:
We have:
[tex]\[ \text{Volume of A} = 2.715 \text{ cubic units} \][/tex]
[tex]\[ \text{Volume of B} = 2.715 \text{ cubic units} \][/tex]
Let's compare the volumes to the given statements:
- Volume [tex]\(B = \frac{1}{2} \)[/tex] Volume [tex]\( A \)[/tex]:
[tex]\[ 2.715 = \frac{1}{2} \times 2.715 \quad \text{(not true)} \][/tex]
- Volume [tex]\(B = \frac{1}{3} \)[/tex] Volume [tex]\( A \)[/tex]:
[tex]\[ 2.715 = \frac{1}{3} \times 2.715 \quad \text{(not true)} \][/tex]
- Volume [tex]\(B = \)[/tex] Volume [tex]\( A \)[/tex]:
[tex]\[ 2.715 = 2.715 \quad \text{(true)} \][/tex]
- Volume [tex]\(B = 2 \times \)[/tex] Volume [tex]\( A \)[/tex]:
[tex]\[ 2.715 = 2 \times 2.715 \quad \text{(not true)} \][/tex]
Therefore, the statement that is true is:
[tex]\[ \text{Volume of B} = \text{Volume of A} \][/tex]
So, the correct statement is:
[tex]\[ \text{Volume } B = \text{Volume } A \][/tex]
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