Question

the base of a solid right pyramid is a square with an edge length of n units. the height of the pyramid is n - 1 units. which expression represents the volume of the pyramid? \\(\frac{1}{3}n^2(n - 1)\\) units³ \\(\frac{1}{3}n(n - 1)\\) units³ \\(\frac{1}{3}n(n - 1)^2\\) units³ \\(\frac{1}{3}n^3(n - 1)\\) units³

Explanation:

Step1: Recall Volume of Pyramid Formula

The volume \( V \) of a pyramid is given by \( V=\frac{1}{3}Bh \), where \( B \) is the area of the base and \( h \) is the height.

Step2: Calculate Base Area

The base is a square with edge length \( n \) units. The area of a square is \( \text{side}^2 \), so \( B = n\times n=n^{2} \) square units.

Step3: Identify Height

The height \( h \) of the pyramid is \( n - 1 \) units.

Step4: Substitute into Volume Formula

Substitute \( B = n^{2} \) and \( h=n - 1 \) into \( V=\frac{1}{3}Bh \). We get \( V=\frac{1}{3}\times n^{2}\times(n - 1)=\frac{1}{3}n^{2}(n - 1) \) cubic units.

Answer:

\(\frac{1}{3}n^{2}(n - 1)\) units\(^{3}\) (the first option)