Answer :
Alright, let's work through this step-by-step to determine the height of each pyramid.
1. Volume of the Cube:
The volume [tex]\( V_{\text{cube}} \)[/tex] of a cube with height [tex]\( h \)[/tex] (which is also the side length) is given by:
[tex]\[ V_{\text{cube}} = h^3 \][/tex]
2. Volume of Each Square Pyramid:
The volume [tex]\( V_{\text{pyramid}} \)[/tex] of a square pyramid with a base side length [tex]\( h \)[/tex] and height [tex]\( h_{\text{pyramid}} \)[/tex] is:
[tex]\[ V_{\text{pyramid}} = \frac{1}{3} \cdot \text{base area} \cdot \text{height} \][/tex]
Since the base area of the pyramid is [tex]\( h^2 \)[/tex] (same as one face of the cube), it becomes:
[tex]\[ V_{\text{pyramid}} = \frac{1}{3} h^2 \cdot h_{\text{pyramid}} \][/tex]
3. Relationship Between Volumes:
According to the problem, six such pyramids can fill the volume of the cube. Hence:
[tex]\[ 6 \cdot V_{\text{pyramid}} = V_{\text{cube}} \][/tex]
Substituting the volumes, we get:
[tex]\[ 6 \cdot \left(\frac{1}{3} h^2 \cdot h_{\text{pyramid}}\right) = h^3 \][/tex]
4. Simplifying the Equation:
[tex]\[ 2 h^2 \cdot h_{\text{pyramid}} = h^3 \][/tex]
Dividing both sides by [tex]\( h^2 \)[/tex]:
[tex]\[ 2 h_{\text{pyramid}} = h \][/tex]
5. Solving for the Height of the Pyramid:
[tex]\[ h_{\text{pyramid}} = \frac{h}{2} \][/tex]
Thus, the height of each pyramid is [tex]\( \frac{h}{3} \)[/tex].
The correct answer is:
[tex]\[ \boxed{\frac{1}{3} h} \][/tex]
1. Volume of the Cube:
The volume [tex]\( V_{\text{cube}} \)[/tex] of a cube with height [tex]\( h \)[/tex] (which is also the side length) is given by:
[tex]\[ V_{\text{cube}} = h^3 \][/tex]
2. Volume of Each Square Pyramid:
The volume [tex]\( V_{\text{pyramid}} \)[/tex] of a square pyramid with a base side length [tex]\( h \)[/tex] and height [tex]\( h_{\text{pyramid}} \)[/tex] is:
[tex]\[ V_{\text{pyramid}} = \frac{1}{3} \cdot \text{base area} \cdot \text{height} \][/tex]
Since the base area of the pyramid is [tex]\( h^2 \)[/tex] (same as one face of the cube), it becomes:
[tex]\[ V_{\text{pyramid}} = \frac{1}{3} h^2 \cdot h_{\text{pyramid}} \][/tex]
3. Relationship Between Volumes:
According to the problem, six such pyramids can fill the volume of the cube. Hence:
[tex]\[ 6 \cdot V_{\text{pyramid}} = V_{\text{cube}} \][/tex]
Substituting the volumes, we get:
[tex]\[ 6 \cdot \left(\frac{1}{3} h^2 \cdot h_{\text{pyramid}}\right) = h^3 \][/tex]
4. Simplifying the Equation:
[tex]\[ 2 h^2 \cdot h_{\text{pyramid}} = h^3 \][/tex]
Dividing both sides by [tex]\( h^2 \)[/tex]:
[tex]\[ 2 h_{\text{pyramid}} = h \][/tex]
5. Solving for the Height of the Pyramid:
[tex]\[ h_{\text{pyramid}} = \frac{h}{2} \][/tex]
Thus, the height of each pyramid is [tex]\( \frac{h}{3} \)[/tex].
The correct answer is:
[tex]\[ \boxed{\frac{1}{3} h} \][/tex]
