Sure, let's consider this problem carefully and systematically.
Firstly, we are dealing with a cube of side length [tex]\( b \)[/tex]. The volume of this cube is given by:
[tex]\[ \text{Volume of the cube} = b \times b \times b = b^3 \][/tex]
Within this cube, four diagonals are drawn, thus creating 6 square pyramids. Each of these pyramids has the same base and height. Since there are 6 pyramids, the volume of one pyramid is:
[tex]\[ \text{Volume of one pyramid} = \frac{1}{6} \text{ of the total volume of the cube} = \frac{b^3}{6} \][/tex]
Now, let's consider the base and height of the pyramids. Each base is a square with side length [tex]\( b \)[/tex], so the base area (B) is:
[tex]\[ B = b \times b = b^2 \][/tex]
The height ([tex]\( h \)[/tex]) given for each pyramid is [tex]\( h \)[/tex]. However, since the height relates to the actual height of the pyramid extending from the base to the apex; herein, each pyramid has a height of [tex]\( 2h \)[/tex]. This corresponds to the diagonal of the cube that bisects the pyramid as per the problem's constraints.
Thus, for each pyramid with a base area [tex]\( B \)[/tex] and a height [tex]\( 2h \)[/tex], the volume can be calculated using the formula for the volume of a pyramid:
[tex]\[ \text{Volume} = \frac{1}{3} \times \text{Base Area} \times \text{Height} \][/tex]
Substituting the values, we have:
[tex]\[ \text{Volume} = \frac{1}{3} \times b^2 \times 2h \][/tex]
Simplifying this, we get:
[tex]\[ \text{Volume} = \frac{2}{3} \times b^2 \times h \][/tex]
Therefore, the correct expression to match the condition specified is:
[tex]\[ \boxed{\frac{2}{3} b^2 h} \][/tex]
To summarize:
1. The volume of the cube is [tex]\( b^3 \)[/tex].
2. The volume of one pyramid, given they are 6 equal pyramids, is [tex]\( \frac{b^3}{6} \)[/tex].
3. The calculated expression for the volume of one pyramid using the traditional volume formula for pyramids is [tex]\( \frac{2}{3} b^2 h \)[/tex].
Hence, the volume of each pyramid matches our required condition provided by the problem. The correct statement is that the volume of one pyramid is [tex]\( \frac{2}{3} B h \)[/tex] where [tex]\( B = b^2 \)[/tex].
