Question

a solid right pyramid has a square base. the base edge length is 2 cm longer than the height of the pyramid. if the height is 6 cm, what is the volume of the pyramid?

\\(24\text{ cm}^3\\)
\\(48\text{ cm}^3\\)
\\(128\text{ cm}^3\\)
\\(144\text{ cm}^3\\)

Explanation:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"tutor",
"mastery_adaptive_brevity"
],
"needs_drawing": false,
"concepts_used": [
"Pyramid Base Geometry",
"Volume of a Right Pyramid"
],
"new_concepts": [],
"current_concepts": [
"Pyramid Base Geometry",
"Volume of a Right Pyramid"
]
}
</pre_analysis>

<reasoning>

Determine the base edge length

Using the Pyramid Base Geometry knowledge point
\[

$$\begin{aligned} h &= 6\text{ cm}\\ s &= h + 2 = 6 + 2 = 8\text{ cm} \end{aligned}$$

\]

Calculate the base area

Using the Pyramid Base Geometry knowledge point
\[
B = s^2 = 8^2 = 64\text{ cm}^2
\]

Calculate the volume of the pyramid

Using the Volume of a Right Pyramid knowledge point
\[
V = \frac{1}{3} \cdot B \cdot h = \frac{1}{3} \cdot 64 \cdot 6 = 128\text{ cm}^3
\]
</reasoning>

<answer>
<mcq-option>(A) \(24\text{ cm}^3\)</mcq-option>
<mcq-option>(B) \(48\text{ cm}^3\)</mcq-option>
<mcq-correct>(C) \(128\text{ cm}^3\)</mcq-correct>
<mcq-option>(D) \(144\text{ cm}^3\)</mcq-option>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Geometry",
"Volume of a Right Pyramid"
]
}
</post_analysis>

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"tutor",
"mastery_adaptive_brevity"
],
"needs_drawing": false,
"concepts_used": [
"Pyramid Base Geometry",
"Volume of a Right Pyramid"
],
"new_concepts": [],
"current_concepts": [
"Pyramid Base Geometry",
"Volume of a Right Pyramid"
]
}
</pre_analysis>

<reasoning>

Determine the base edge length

Using the Pyramid Base Geometry knowledge point
\[

$$\begin{aligned} h &= 6\text{ cm}\\ s &= h + 2 = 6 + 2 = 8\text{ cm} \end{aligned}$$

\]

Calculate the base area

Using the Pyramid Base Geometry knowledge point
\[
B = s^2 = 8^2 = 64\text{ cm}^2
\]

Calculate the volume of the pyramid

Using the Volume of a Right Pyramid knowledge point
\[
V = \frac{1}{3} \cdot B \cdot h = \frac{1}{3} \cdot 64 \cdot 6 = 128\text{ cm}^3
\]
</reasoning>

<answer>
<mcq-option>(A) \(24\text{ cm}^3\)</mcq-option>
<mcq-option>(B) \(48\text{ cm}^3\)</mcq-option>
<mcq-correct>(C) \(128\text{ cm}^3\)</mcq-correct>
<mcq-option>(D) \(144\text{ cm}^3\)</mcq-option>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Geometry",
"Volume of a Right Pyramid"
]
}
</post_analysis>