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(n - 1)\text{ units}^3\\)
\\(\frac{1}{3}n(n - 1)^2\text{ units}^3\\)
\\(\frac{1}{3}n^2(n - 1)\text{ units}^3\\)
\\(\frac{1}{3}n^3(n - 1)\text{ units}^3\\)

Explanation:

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<reasoning>

Calculate the base area of the square pyramid

\[
B = n^2
\]

Apply the pyramid volume formula

\[

$$\begin{aligned} V &= \frac{1}{3} \cdot B \cdot h \\ V &= \frac{1}{3} n^2 (n - 1) \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(\frac{1}{3}n(n - 1)\text{ units}^3\)</mcq-option>
<mcq-option>(B) \(\frac{1}{3}n(n - 1)^2\text{ units}^3\)</mcq-option>
<mcq-correct>(C) \(\frac{1}{3}n^2(n - 1)\text{ units}^3\)</mcq-correct>
<mcq-option>(D) \(\frac{1}{3}n^3(n - 1)\text{ units}^3\)</mcq-option>
</answer>

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Answer:

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<reasoning>

Calculate the base area of the square pyramid

\[
B = n^2
\]

Apply the pyramid volume formula

\[

$$\begin{aligned} V &= \frac{1}{3} \cdot B \cdot h \\ V &= \frac{1}{3} n^2 (n - 1) \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(\frac{1}{3}n(n - 1)\text{ units}^3\)</mcq-option>
<mcq-option>(B) \(\frac{1}{3}n(n - 1)^2\text{ units}^3\)</mcq-option>
<mcq-correct>(C) \(\frac{1}{3}n^2(n - 1)\text{ units}^3\)</mcq-correct>
<mcq-option>(D) \(\frac{1}{3}n^3(n - 1)\text{ units}^3\)</mcq-option>
</answer>

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