IDNLearn.com is designed to help you find reliable answers quickly and easily. Get comprehensive and trustworthy answers to all your questions from our knowledgeable community members.
Sagot :
To determine the volume of one of the six square pyramids formed within a cube, let's break down the problem step by step.
1. Volume of the Cube:
The volume [tex]\( V \)[/tex] of a cube with side length [tex]\( b \)[/tex] is given by:
[tex]\[ V_{\text{cube}} = b^3 \][/tex]
2. Understanding the Pyramids:
Inside the cube, when the four diagonals are drawn, it creates six square pyramids. Each pyramid has its base as one of the square faces of the cube and a height equal to half the body diagonal of the cube.
3. Height of the Pyramids:
For each of these pyramids, the height can be considered as the body diagonal of the cube divided by 2. Given that the height [tex]\( h \)[/tex] provided corresponds with the cube's side, we have:
[tex]\[ h = \frac{\sqrt{3}}{2}b \][/tex]
4. Volume of a Pyramid:
The volume [tex]\( V \)[/tex] of a pyramid with a square base is given by:
[tex]\[ V_{\text{pyramid}} = \frac{1}{3} \times [\text{Base Area}] \times [\text{Height}] \][/tex]
The base area of the pyramid is:
[tex]\[ \text{Base Area} = b^2 \][/tex]
Substituting the height [tex]\( h \)[/tex] as:
[tex]\[ V_{\text{pyramid}} = \frac{1}{3} \times b^2 \times 2h \][/tex]
Simplifying:
[tex]\[ V_{\text{pyramid}} = \frac{1}{3} \times b^2 \times 2 \times \frac{1}{2} = \frac{1}{6} \times b^2 \times b \times 2 = \frac{1}{6}(b^2)(2h) \][/tex]
5. Comparing the Options:
We need to determine which expression from the options matches our derived formula:
[tex]\[ \frac{1}{6}(b^2)(2h) = \frac{1}{3}b^2h \][/tex]
We compare it to the options given:
- [tex]\(\frac{1}{6}(b)(b)(2h)\)[/tex] or [tex]\(\frac{1}{3}bh\)[/tex]
- [tex]\(\frac{1}{6}(b)(b)(6h)\)[/tex] or [tex]\(bh\)[/tex]
- [tex]\(\frac{1}{3}(b)(b)(6h)\)[/tex] or [tex]\(2bh\)[/tex]
- [tex]\(\frac{1}{3}(b)(b)(2h)\)[/tex] or [tex]\(\frac{2}{3}bh\)[/tex]
The correct one that matches our derived volume is:
[tex]\[ \frac{1}{6}(b^2)(2h) \][/tex]
or equivalently:
[tex]\[ \frac{1}{3}b^2h \][/tex]
Therefore, the correct choice is:
[tex]\[ \frac{1}{6}(b)(b)(2h) \text{ or } \frac{1}{3}bh. \][/tex]
1. Volume of the Cube:
The volume [tex]\( V \)[/tex] of a cube with side length [tex]\( b \)[/tex] is given by:
[tex]\[ V_{\text{cube}} = b^3 \][/tex]
2. Understanding the Pyramids:
Inside the cube, when the four diagonals are drawn, it creates six square pyramids. Each pyramid has its base as one of the square faces of the cube and a height equal to half the body diagonal of the cube.
3. Height of the Pyramids:
For each of these pyramids, the height can be considered as the body diagonal of the cube divided by 2. Given that the height [tex]\( h \)[/tex] provided corresponds with the cube's side, we have:
[tex]\[ h = \frac{\sqrt{3}}{2}b \][/tex]
4. Volume of a Pyramid:
The volume [tex]\( V \)[/tex] of a pyramid with a square base is given by:
[tex]\[ V_{\text{pyramid}} = \frac{1}{3} \times [\text{Base Area}] \times [\text{Height}] \][/tex]
The base area of the pyramid is:
[tex]\[ \text{Base Area} = b^2 \][/tex]
Substituting the height [tex]\( h \)[/tex] as:
[tex]\[ V_{\text{pyramid}} = \frac{1}{3} \times b^2 \times 2h \][/tex]
Simplifying:
[tex]\[ V_{\text{pyramid}} = \frac{1}{3} \times b^2 \times 2 \times \frac{1}{2} = \frac{1}{6} \times b^2 \times b \times 2 = \frac{1}{6}(b^2)(2h) \][/tex]
5. Comparing the Options:
We need to determine which expression from the options matches our derived formula:
[tex]\[ \frac{1}{6}(b^2)(2h) = \frac{1}{3}b^2h \][/tex]
We compare it to the options given:
- [tex]\(\frac{1}{6}(b)(b)(2h)\)[/tex] or [tex]\(\frac{1}{3}bh\)[/tex]
- [tex]\(\frac{1}{6}(b)(b)(6h)\)[/tex] or [tex]\(bh\)[/tex]
- [tex]\(\frac{1}{3}(b)(b)(6h)\)[/tex] or [tex]\(2bh\)[/tex]
- [tex]\(\frac{1}{3}(b)(b)(2h)\)[/tex] or [tex]\(\frac{2}{3}bh\)[/tex]
The correct one that matches our derived volume is:
[tex]\[ \frac{1}{6}(b^2)(2h) \][/tex]
or equivalently:
[tex]\[ \frac{1}{3}b^2h \][/tex]
Therefore, the correct choice is:
[tex]\[ \frac{1}{6}(b)(b)(2h) \text{ or } \frac{1}{3}bh. \][/tex]
Thank you for being part of this discussion. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Thank you for choosing IDNLearn.com. We’re dedicated to providing clear answers, so visit us again for more solutions.