To solve the problem, let's break it down step-by-step.
1. Understanding the Problem:
The composite figure consists of two identical pyramids attached at their bases. Each pyramid has a height of 2 units. You are given a few expressions to choose from, which represent the volume of the composite figure.
2. Volume of a Pyramid:
The formula for the volume of a pyramid is given by:
[tex]\[
V = \frac{1}{3} \times (\text{base area}) \times (\text{height})
\][/tex]
3. Given Data:
- Each pyramid has a height of 2 units.
- The base area is given as a factor between [tex]\(5\)[/tex] and [tex]\(0.25\)[/tex].
4. Calculate the Volume of One Pyramid:
Substituting the given values into the volume formula, we have:
[tex]\[
\text{Volume of one pyramid} = \frac{1}{3} \times 5 \times 0.25 \times 2
\][/tex]
5. Evaluating the Volume of One Pyramid:
Let's simplify the calculation:
- First, calculate the base area multiplied by one third and multiplied by height.
- [tex]\[
\text{Volume of one pyramid} = \frac{1}{3} \times 5 \times 0.25 \times 2 = \frac{1}{3} \times 1.25 \times 2 = \frac{1}{3} \times 2.5 = 0.833333\ldots
\approx 0.833
\][/tex]
6. Volume of Composite Figure:
The composite figure is made up of two such pyramids. Therefore, the total volume is twice the volume of one pyramid:
[tex]\[
\text{Volume of the composite figure} = 2 \times 0.833 = 1.666666\ldots \approx 1.667
\][/tex]
Now, let’s match our calculation to the given expressions:
- First option:
[tex]\(\frac{1}{2}\left(\frac{1}{3}(5)(0.25)(2)\right)\)[/tex]
- This results in half the volume of one pyramid, not the total volume of the composite figure.
- Second option:
[tex]\(\frac{1}{2}\left(\frac{1}{3}(5)(0.25)(4)\right)\)[/tex]
- This involves doubling the height in the base volume, which would incorrectly calculate the volume.
- Third option:
[tex]\(2\left(\frac{1}{3}(5)(0.25)(2)\right)\)[/tex]
- This correctly represents twice the volume of one pyramid, aligning with our correct calculation.
- Fourth option:
[tex]\(2\left(\frac{1}{3}(5)(0.25)(4)\right)\)[/tex]
- This would incorrectly calculate the volume by doubling the height.
Thus, the correct expression is:
[tex]\[
2\left(\frac{1}{3}(5)(0.25)(2)\right)
\][/tex]
This expression represents the volume, in cubic units, of the composite figure, which is approximately [tex]\(1.667\)[/tex] cubic units.
