To find the volume of the composite figure, which is made up of two identical pyramids attached at their bases, we will follow these steps:
1. Determine the base area and height of one pyramid:
- Each pyramid has a height of [tex]\(2\)[/tex] units.
- The base area of each pyramid is given by multiplying the base length by its width. Since it gives [tex]\(5 \times 0.25\)[/tex] square units, the base area is [tex]\(1.25\)[/tex] square units.
2. Calculate the volume of one pyramid:
- The formula for the volume [tex]\(V\)[/tex] of a pyramid is:
[tex]\[
V = \frac{1}{3} \times \text{Base Area} \times \text{Height}
\][/tex]
- Substitute the known values:
[tex]\[
V = \frac{1}{3} \times 1.25 \times 2 = 0.8333333333333333 \, \text{cubic units}
\][/tex]
3. Determine the volume of the entire composite figure:
- There are two identical pyramids, so the total volume is:
[tex]\[
\text{Total Volume} = 2 \times V = 2 \times 0.8333333333333333 = 1.6666666666666665 \, \text{cubic units}
\][/tex]
4. Analyze each given expression to find which correctly represents the volume of the composite figure:
- [tex]\(\frac{1}{2}\left(\frac{1}{3}(5)(0.25)(2)\right)\)[/tex]:
[tex]\[
= \frac{1}{2} \left(\frac{1}{3} \times 5 \times 0.25 \times 2\right)
= \frac{1}{2} \times 0.8333333333333333
= 0.41666666666666663 \, \text{cubic units}
\][/tex]
- [tex]\(\frac{1}{2}\left(\frac{1}{3}(5)(0.25)(4)\right)\)[/tex]:
[tex]\[
= \frac{1}{2} \left(\frac{1}{3} \times 5 \times 0.25 \times 4\right)
= \frac{1}{2} \times 1.6666666666666665
= 0.8333333333333333 \, \text{cubic units}
\][/tex]
- [tex]\(2\left(\frac{1}{3}(5)(0.25)(2)\right)\)[/tex]:
[tex]\[
= 2 \left(\frac{1}{3} \times 5 \times 0.25 \times 2\right)
= 2 \times 0.8333333333333333
= 1.6666666666666665 \, \text{cubic units}
\][/tex]
- [tex]\(2\left(\frac{1}{3}(5)(0.25)(4)\right)\)[/tex]:
[tex]\[
= 2 \left(\frac{1}{3} \times 5 \times 0.25 \times 4\right)
= 2 \times 1.6666666666666665
= 3.333333333333333 \, \text{cubic units}
\][/tex]
Based on our analysis, the expression that correctly represents the volume of the composite figure is:
[tex]\[
2\left(\frac{1}{3}(5)(0.25)(2)\right)
\][/tex]
This is the expression that matches the total volume of [tex]\(1.6666666666666665\)[/tex] cubic units.
