Answer :
To understand the height of each square pyramid, we need to consider how these pyramids relate to the cube they fill. Let's break it down step-by-step:
1. Volume of the Cube:
- The volume of a cube with a height (which is also the length of its sides) [tex]\( h \)[/tex] units can be calculated using the formula for the volume of a cube: [tex]\( V_{\text{cube}} = h^3 \)[/tex].
2. Volume of Each Pyramid:
- According to the problem, six identical square pyramids fill the same volume as the cube. Therefore, the volume of one pyramid is one-sixth of the volume of the cube:
[tex]\[ V_{\text{pyramid}} = \frac{V_{\text{cube}}}{6} = \frac{h^3}{6} \][/tex]
3. Volume Formula for a Square Pyramid:
- The volume [tex]\( V \)[/tex] of a square pyramid is given by the formula:
[tex]\[ V_{\text{pyramid}} = \frac{1}{3} \times \text{base area} \times \text{height} \][/tex]
- For a square pyramid with the same base as the cube, the base area is [tex]\( h^2 \)[/tex]. Let [tex]\( h_{\text{pyramid}} \)[/tex] be the height of the pyramid.
4. Setting Up the Equation:
- Substitute the volume of the pyramid and the base area into the volume formula:
[tex]\[ \frac{1}{3} \times h^2 \times h_{\text{pyramid}} = \frac{h^3}{6} \][/tex]
5. Solving for the Height [tex]\( h_{\text{pyramid}} \)[/tex]:
- To find [tex]\( h_{\text{pyramid}} \)[/tex], let's solve the equation:
[tex]\[ \frac{1}{3} h^2 \times h_{\text{pyramid}} = \frac{h^3}{6} \][/tex]
- Multiply both sides by 3 to clear the fraction on the left side:
[tex]\[ h^2 \times h_{\text{pyramid}} = \frac{h^3}{2} \][/tex]
- Divide both sides by [tex]\( h^2 \)[/tex]:
[tex]\[ h_{\text{pyramid}} = \frac{h^3}{2h^2} = \frac{h}{2} \][/tex]
Therefore, the height of each pyramid is [tex]\( \frac{1}{2} h \)[/tex] units.
Among the given options, the correct answer is:
- The height of each pyramid is [tex]\( \frac{1}{2} h \)[/tex] units.
1. Volume of the Cube:
- The volume of a cube with a height (which is also the length of its sides) [tex]\( h \)[/tex] units can be calculated using the formula for the volume of a cube: [tex]\( V_{\text{cube}} = h^3 \)[/tex].
2. Volume of Each Pyramid:
- According to the problem, six identical square pyramids fill the same volume as the cube. Therefore, the volume of one pyramid is one-sixth of the volume of the cube:
[tex]\[ V_{\text{pyramid}} = \frac{V_{\text{cube}}}{6} = \frac{h^3}{6} \][/tex]
3. Volume Formula for a Square Pyramid:
- The volume [tex]\( V \)[/tex] of a square pyramid is given by the formula:
[tex]\[ V_{\text{pyramid}} = \frac{1}{3} \times \text{base area} \times \text{height} \][/tex]
- For a square pyramid with the same base as the cube, the base area is [tex]\( h^2 \)[/tex]. Let [tex]\( h_{\text{pyramid}} \)[/tex] be the height of the pyramid.
4. Setting Up the Equation:
- Substitute the volume of the pyramid and the base area into the volume formula:
[tex]\[ \frac{1}{3} \times h^2 \times h_{\text{pyramid}} = \frac{h^3}{6} \][/tex]
5. Solving for the Height [tex]\( h_{\text{pyramid}} \)[/tex]:
- To find [tex]\( h_{\text{pyramid}} \)[/tex], let's solve the equation:
[tex]\[ \frac{1}{3} h^2 \times h_{\text{pyramid}} = \frac{h^3}{6} \][/tex]
- Multiply both sides by 3 to clear the fraction on the left side:
[tex]\[ h^2 \times h_{\text{pyramid}} = \frac{h^3}{2} \][/tex]
- Divide both sides by [tex]\( h^2 \)[/tex]:
[tex]\[ h_{\text{pyramid}} = \frac{h^3}{2h^2} = \frac{h}{2} \][/tex]
Therefore, the height of each pyramid is [tex]\( \frac{1}{2} h \)[/tex] units.
Among the given options, the correct answer is:
- The height of each pyramid is [tex]\( \frac{1}{2} h \)[/tex] units.
