Find answers to your questions faster and easier with IDNLearn.com. Discover in-depth and trustworthy answers from our extensive network of knowledgeable professionals.
Sagot :
Let's walk through the problem step-by-step to understand the relationship between the height of the cube and the height of the pyramids that fill it.
1. Volume of the Cube:
- Given the height [tex]\( h \)[/tex] of the cube, we know that the side length of the cube is also [tex]\( h \)[/tex] because all sides of a cube are equal.
- The volume [tex]\( V_{\text{cube}} \)[/tex] of the cube is calculated as:
[tex]\[ V_{\text{cube}} = h^3 \][/tex]
2. Volume of a Square Pyramid:
- The base area [tex]\( A_{\text{base}} \)[/tex] of the square pyramid is the same as one face of the cube, so:
[tex]\[ A_{\text{base}} = h^2 \][/tex]
- Let the height of the pyramid be [tex]\( h_{\text{pyramid}} \)[/tex].
- The volume [tex]\( V_{\text{pyramid}} \)[/tex] of a square pyramid is given by:
[tex]\[ V_{\text{pyramid}} = \frac{1}{3} \times \text{Base Area} \times \text{Height} = \frac{1}{3} \times h^2 \times h_{\text{pyramid}} \][/tex]
3. Relationship Between the Cubic and Pyramidal Volumes:
- We are given that six identical square pyramids perfectly fill the volume of the cube. Therefore:
[tex]\[ 6 \times V_{\text{pyramid}} = V_{\text{cube}} \][/tex]
- Substitute the volumes into this equation:
[tex]\[ 6 \times \left( \frac{1}{3} \times h^2 \times h_{\text{pyramid}} \right) = h^3 \][/tex]
- Simplify the equation:
[tex]\[ 2 \times h^2 \times h_{\text{pyramid}} = h^3 \][/tex]
4. Solve for the Height of Each Pyramid:
- To find [tex]\( h_{\text{pyramid}} \)[/tex], divide both sides of the equation by [tex]\( 2h^2 \)[/tex]:
[tex]\[ h_{\text{pyramid}} = \frac{h^3}{2h^2} = \frac{h}{2} \][/tex]
Therefore, the height of each pyramid is:
[tex]\[ h_{\text{pyramid}} = \frac{1}{2} h \][/tex]
The correct answer is:
- The height of each pyramid is [tex]\(\frac{1}{2} h\)[/tex] units.
1. Volume of the Cube:
- Given the height [tex]\( h \)[/tex] of the cube, we know that the side length of the cube is also [tex]\( h \)[/tex] because all sides of a cube are equal.
- The volume [tex]\( V_{\text{cube}} \)[/tex] of the cube is calculated as:
[tex]\[ V_{\text{cube}} = h^3 \][/tex]
2. Volume of a Square Pyramid:
- The base area [tex]\( A_{\text{base}} \)[/tex] of the square pyramid is the same as one face of the cube, so:
[tex]\[ A_{\text{base}} = h^2 \][/tex]
- Let the height of the pyramid be [tex]\( h_{\text{pyramid}} \)[/tex].
- The volume [tex]\( V_{\text{pyramid}} \)[/tex] of a square pyramid is given by:
[tex]\[ V_{\text{pyramid}} = \frac{1}{3} \times \text{Base Area} \times \text{Height} = \frac{1}{3} \times h^2 \times h_{\text{pyramid}} \][/tex]
3. Relationship Between the Cubic and Pyramidal Volumes:
- We are given that six identical square pyramids perfectly fill the volume of the cube. Therefore:
[tex]\[ 6 \times V_{\text{pyramid}} = V_{\text{cube}} \][/tex]
- Substitute the volumes into this equation:
[tex]\[ 6 \times \left( \frac{1}{3} \times h^2 \times h_{\text{pyramid}} \right) = h^3 \][/tex]
- Simplify the equation:
[tex]\[ 2 \times h^2 \times h_{\text{pyramid}} = h^3 \][/tex]
4. Solve for the Height of Each Pyramid:
- To find [tex]\( h_{\text{pyramid}} \)[/tex], divide both sides of the equation by [tex]\( 2h^2 \)[/tex]:
[tex]\[ h_{\text{pyramid}} = \frac{h^3}{2h^2} = \frac{h}{2} \][/tex]
Therefore, the height of each pyramid is:
[tex]\[ h_{\text{pyramid}} = \frac{1}{2} h \][/tex]
The correct answer is:
- The height of each pyramid is [tex]\(\frac{1}{2} h\)[/tex] units.
We appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. Thank you for choosing IDNLearn.com for your queries. We’re here to provide accurate answers, so visit us again soon.