To determine the height of each pyramid, we need to use the information given: Six identical square pyramids can fill the same volume as a cube with the same base. Let’s proceed step-by-step to find the correct height of each pyramid.
1. Volume of the Cube:
- We are given that the height of the cube is [tex]\( h \)[/tex] units.
- The volume [tex]\( V_{\text{cube}} \)[/tex] of the cube is calculated as [tex]\( h^3 \)[/tex] since the length, width, and height of the cube are all [tex]\( h \)[/tex].
2. Volume of Each Pyramid:
- The volume [tex]\( V_{\text{pyramid}} \)[/tex] of a pyramid is given by the formula:
[tex]\[
V_{\text{pyramid}} = \frac{1}{3} \times \text{base area} \times \text{height of the pyramid}
\][/tex]
- For our pyramids, the base area of each pyramid is the same as the base area of the cube, which is [tex]\( h^2 \)[/tex].
3. Total Volume of Six Pyramids:
- Since six identical pyramids fill the volume of the cube, we can write:
[tex]\[
6 \times V_{\text{pyramid}} = V_{\text{cube}}
\][/tex]
- Substituting the volume expressions, we get:
[tex]\[
6 \times \left( \frac{1}{3} \times h^2 \times \text{height of the pyramid} \right) = h^3
\][/tex]
4. Solving for the Height of Each Pyramid:
- Simplifying the equation:
[tex]\[
6 \times \left( \frac{1}{3} \times h^2 \times \text{height of the pyramid} \right) = h^3
\][/tex]
[tex]\[
2 \times h^2 \times \text{height of the pyramid} = h^3
\][/tex]
- Dividing both sides by [tex]\( 2h^2 \)[/tex]:
[tex]\[
\text{height of the pyramid} = \frac{h^3}{2h^2}
\][/tex]
[tex]\[
\text{height of the pyramid} = \frac{h}{2}
\][/tex]
Thus, the height of each pyramid is [tex]\( \frac{h}{2} \)[/tex] units.
Given the options:
- The height of each pyramid is [tex]\( \frac{1}{2} h \)[/tex] units.
- The height of each pyramid is [tex]\( \frac{1}{3} h \)[/tex] units.
- The height of each pyramid is [tex]\( \frac{1}{6} h \)[/tex] units.
- The height of each pyramid is [tex]\( h \)[/tex] units.
The correct answer is: The height of each pyramid is [tex]\( \frac{1}{2} h \)[/tex] units.
