Question

the four diagonals of a cube are drawn to create 6 square pyramids with the same base and height. the volume of the cube is (b)(b)(b). the height of each pyramid is h. therefore, the volume of one pyramid must equal one - sixth the volume of the cube, or
○ $\frac{1}{6}(b)(b)(2h)$ or $\frac{1}{3}bh$.
○ $\frac{1}{6}(b)(b)(6h)$ or $bh$.
○ $\frac{1}{3}(b)(b)(6h)$ or $\frac{1}{3}bh$.
○ $\frac{1}{3}(b)(b)(2h)$ or $\frac{2}{3}bh$.

Explanation:

Step1: Recall volume of cube and pyramid

Volume of cube \(V_{cube}=b\times b\times b = b^{3}\). Volume of a square - pyramid \(V_{pyramid}=\frac{1}{3}Bh\), where \(B\) is the base - area and \(h\) is the height. For a square - based pyramid formed from a cube, the base of the pyramid is a face of the cube, so \(B = b\times b\) and the height of the pyramid \(h=b\). Since 6 such pyramids make up a cube, \(V_{pyramid}=\frac{1}{6}V_{cube}\).

Step2: Calculate volume of one pyramid

\(V_{pyramid}=\frac{1}{6}(b)(b)(b)\). Also, since \(B = b\times b\) and \(h = b\), \(V_{pyramid}=\frac{1}{3}Bh\) where \(B=b^{2}\) and \(h = b\). The volume of one pyramid is \(\frac{1}{6}(b)(b)(b)\) or \(\frac{1}{3}Bh\) (because \(B = b^{2}\) and \(h = b\)).

Answer:

The volume of one pyramid is \(\frac{1}{6}(b)(b)(b)\) or \(\frac{1}{3}Bh\) (corresponding to the correct option among the given ones which is not fully clear from the text but based on the analysis this is the correct relationship). If we assume the options are written in a more standard form, the correct option is the one that shows \(\frac{1}{6}(b)(b)(b)\) or \(\frac{1}{3}Bh\) with \(B = b^{2}\) and \(h = b\).