Question

a sculptor is designing a pyramid - shaped model for a museum exhibit. the base of the pyramid is a square with a side length of 6 meters, and its height is 9 meters. to estimate the material needed, the sculptor calculates the volume of the pyramid. the sculptor also notes that if a rectangular prism were built with the same base and height, its volume would be three times greater than the pyramids. which of the following statements correctly describes the relationship between the volume of the pyramid and the prism? the volume of the pyramid is three times the volume of the prism. the volume of the prism is half the volume of the pyramid. the volume of the prism and the pyramid are equal. the volume of the pyramid is one - third the volume of the prism.

Explanation:

Step1: Recall volume formulas

The volume formula for a pyramid is \(V_{pyramid}=\frac{1}{3}Bh\), where \(B\) is the base area and \(h\) is the height. For a square - based pyramid with side length \(s = 6\) meters, \(B=s^{2}=6^{2}=36\) square meters and \(h = 9\) meters. So \(V_{pyramid}=\frac{1}{3}\times36\times9\).
The volume formula for a rectangular prism is \(V_{prism}=Bh\), with the same \(B = 36\) square meters and \(h = 9\) meters. So \(V_{prism}=36\times9\).

Step2: Compare the two volumes

Let's calculate the ratio \(\frac{V_{pyramid}}{V_{prism}}\). Substitute the formulas: \(\frac{V_{pyramid}}{V_{prism}}=\frac{\frac{1}{3}Bh}{Bh}\). Since \(B\) and \(h\) are non - zero (in this context, \(B = 36
eq0\) and \(h=9
eq0\)), the \(B\) and \(h\) terms cancel out. So \(\frac{V_{pyramid}}{V_{prism}}=\frac{1}{3}\), which means \(V_{pyramid}=\frac{1}{3}V_{prism}\).

Answer:

The volume of the pyramid is one - third the volume of the prism.